2000 character limit reached
On convergence of waveform relaxation for nonlinear systems of ordinary differential equations (2307.00276v2)
Published 1 Jul 2023 in math.NA, cs.CE, cs.NA, and physics.comp-ph
Abstract: To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard-Lindel\"of iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, three-dimensional Liouville-Bratu-Gelfand, and three-dimensional nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators.
- H.C. Elman, D.J. Silvester, A.J. Wathen, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics (Oxford University Press, Oxford, 2005) [3] M. Hochbruck, A. Ostermann, Exponential integrators. Acta Numer. 19, 209–286 (2010). 10.1017/S0962492910000048 [4] P.N. Brown, Y. Saad, Convergence theory of nonlinear Newton–Krylov algorithms. SIAM J. Optimization 4(2), 297–330 (1994) [5] R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, A. Ostermann, Exponential integrators. Acta Numer. 19, 209–286 (2010). 10.1017/S0962492910000048 [4] P.N. Brown, Y. Saad, Convergence theory of nonlinear Newton–Krylov algorithms. SIAM J. Optimization 4(2), 297–330 (1994) [5] R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.N. Brown, Y. Saad, Convergence theory of nonlinear Newton–Krylov algorithms. SIAM J. Optimization 4(2), 297–330 (1994) [5] R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.N. Brown, Y. Saad, Convergence theory of nonlinear Newton–Krylov algorithms. SIAM J. Optimization 4(2), 297–330 (1994) [5] R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. 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Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R. Choquet, J. Erhel, Newton–GMRES algorithm applied to compressible flows. Int. J. Numer. Meth. Fluids 23, 177–190 (1996) [6] D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Tromeur-Dervout, Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Journal of Computational Physics 219(1), 210–227 (2006). https://doi.org/10.1016/j.jcp.2006.03.014. URL https://www.sciencedirect.com/science/article/pii/S0021999106001483 [7] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables (Springer-Verlag, New York, 1971), pp. viii+160 [8] E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- E.G. D’yakonov, Difference systems of second order accuracy with a divided operator for parabolic equations without mixed derivatives. USSR Comput. Math. Math. Phys. 4(5), 206–216 (1964) [9] P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P. Csomós, I. Faragó, A. Havasi, Weighted sequential splittings and their analysis. Comput. Math. Appl. pp. 1017–1031 (2005) [10] J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. 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Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. 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Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, E.J. Spee, J.G. Blom, W. Hundsdorfer, A second order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Sci. Comput. 20, 456–480 (1999) [11] P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) P.J. van der Houwen, B.P. Sommeijer, On the internal stability of explicit m𝑚mitalic_m-stage Runge–Kutta methods for large values of m𝑚mitalic_m. Z. Angew. Math. Mech. 60, 479–485 (1980) [12] V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.O. Lokutsievskii, O.V. Lokutsievskii, On numerical solution of boundary value problems for equations of parabolic type. Soviet Math. Dokl. 34(3), 512–516 (1987) [13] B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. 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BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: An explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- J. Comput. Appl. Math. 88, 315–326 (1998) [14] H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- H. Tal-Ezer, Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26(1), 1–11 (1989) [15] V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- V.I. Lebedev, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum. Russian J. Numer. Anal. Math. Modelling 13(2), 107–116 (1998). 10.1515/rnam.1998.13.2.107 [16] J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J.G. Verwer, B.P. Sommeijer, W. Hundsdorfer, RKC time-stepping for advection–diffusion–reaction problems. Journal of Computational Physics 201(1), 61–79 (2004). https://doi.org/10.1016/j.jcp.2004.05.002. URL https://www.sciencedirect.com/science/article/pii/S0021999104001925 [17] M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, G.L.G. Sleijpen, H.A. van der Vorst, Stability control for approximate implicit time stepping schemes with minimum residual iterations. Appl. Numer. Math. 31(3), 239–253 (1999). https://doi.org/10.1016/S0168-9274(98)00138-X [18] M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, H.A. van der Vorst, A parallel nearly implicit scheme. Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Journal of Computational and Applied Mathematics 137, 229–243 (2001). https://doi.org/10.1016/S0377-0427(01)00358-2 [19] V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, Explicit methods of numerical integration for parabolic equations. Mathematical Models and Computer Simulations 3(3), 311–332 (2011). https://doi.org/10.1134/S2070048211030136 [20] E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Lelarasmee, A.E. Ruehli, A.L. Sangiovanni-Vincentelli, The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 1(3), 131–145 (1982). 10.1109/TCAD.1982.1270004 [21] A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) A.R. Newton, A.L. Sangiovanni-Vincentelli, Relaxation-based electrical simulation. IEEE Transactions on Electron Devices 30(9), 1184–1207 (1983). 10.1109/T-ED.1983.21275 [22] S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. 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Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) S. Vandewalle, Parallel Multigrid Waveform Relaxation for Parabolic Problems (Teubner, Stuttgart, 1993) [23] U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- U. Miekkala, O. Nevanlinna, Iterative solution of systems of linear differential equations. Acta Numerica 5, 259–307 (1996). https://doi.org/10.1017/S096249290000266X [24] J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. White, F. Odeh, A.L. Sangiovanni-Vincentelli, A. Ruehli, Waveform relaxation: Theory and practice. Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Tech. Rep. UCB/ERL M85/65, EECS Department, University of California, Berkeley (1985). www.eecs.berkeley.edu/Pubs/TechRpts/1985/543.html [25] U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. 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Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) U. Miekkala, O. Nevanlinna, Convergence of dynamic iteration methods for initial value problems. SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM Journal on Scientific and Statistical Computing 8(4), 459–482 (1987). https://doi.org/10.1137/0908046 [26] C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) C. Lubich, A. Ostermann, Multi-grid dynamic iteration for parabolic equations. BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- BIT Numerical Mathematics 27, 216–234 (1987). URL http://doi.org/10.1007/BF01934186. 10.1007/BF01934186 [27] J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Janssen, S. Vandewalle, Multigrid waveform relaxation of spatial finite element meshes: The continuous-time case. SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Numer. Anal. 33(2), 456–474 (1996). 10.1137/0733024 [28] M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.J. Gander, S. Güttel, PARAEXP: A parallel integrator for linear initial-value problems. SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM Journal on Scientific Computing 35(2), C123–C142 (2013) [29] G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G.L. Kooij, M.A. Botchev, B.J. Geurts, A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations. Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Journal of Computational and Applied Mathematics 316(Supplement C), 229–246 (2017). https://doi.org/10.1016/j.cam.2016.09.036 [30] G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) G. Kooij, M.A. Botchev, B.J. Geurts, An exponential time integrator for the incompressible Navier–Stokes equation. SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Sci. Comput. 40(3), B684–B705 (2018). https://doi.org/10.1137/17M1121950 [31] T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.J. Park, J.C. Light, Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- J. Chem. Phys. 85, 5870–5876 (1986) [32] H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) H.A. van der Vorst, An iterative solution method for solving f(A)x=b𝑓𝐴𝑥𝑏f(A)x=bitalic_f ( italic_A ) italic_x = italic_b, using Krylov subspace information obtained for the symmetric positive definite matrix A𝐴Aitalic_A. J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- J. Comput. Appl. Math. 18, 249–263 (1987) [33] V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices. U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- U.S.S.R. Comput. Maths. Math. Phys. 29(6), 112–121 (1989) [34] E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Gallopoulos, Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992). http://doi.org/10.1137/0913071 [35] V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.L. Druskin, L.A. Knizhnerman, Krylov subspace approximations of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Lin. Alg. Appl. 2, 205–217 (1995) [36] M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Numer. Anal. 34(5), 1911–1925 (1997) [37] M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- M.A. Botchev, A block Krylov subspace time-exact solution method for linear ordinary differential equation systems. Numer. Linear Algebra Appl. 20(4), 557–574 (2013). http://doi.org/10.1002/nla.1865 [38] O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) O. Nevanlinna, F. Odeh, Remarks on the convergence of waveform relaxation method. Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Numerical functional analysis and optimization 9(3-4), 435–445 (1987). https://doi.org/10.1080/01630568708816241 [39] K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) K. Dekker, J.G. Verwer, Stability of Runge–Kutta methods for stiff non-linear differential equations (North-Holland Elsevier Science Publishers, 1984) [40] M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- M.A. Botchev, V. Grimm, M. Hochbruck, Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013). http://doi.org/10.1137/110820191 [41] M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, I.V. Oseledets, E.E. Tyrtyshnikov, Iterative across-time solution of linear differential equations: Krylov subspace versus waveform relaxation. Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Computers & Mathematics with Applications 67(12), 2088–2098 (2014). http://doi.org/10.1016/j.camwa.2014.03.002 [42] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential–Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Springer Series in Computational Mathematics 14 (Springer–Verlag, 1996) [43] L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- L.A. Krukier, Implicit difference schemes and an iterative method for solving them for a certain class of systems of quasi-linear equations. Sov. Math. 23(7), 43–55 (1979). Translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1979, No. 7(206), 41–52 (1979) [44] R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) R.W.C.P. Verstappen, A.E.P. Veldman, Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- J. Comput. Phys. 187(1), 343–368 (2003). http://doi.org/10.1016/S0021-9991(03)00126-8 [45] I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) I. Moret, P. Novati, RD rational approximations of the matrix exponential. BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- BIT 44, 595–615 (2004) [46] J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. van den Eshof, M. Hochbruck, Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Sci. Comput. 27(4), 1438–1457 (2006) [47] L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- SIAM J. Sci. Comput. 18(1), 1–22 (1997). Available at www.mathworks.com [48] J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) J. Jon, S. Klaus, The Liouville–Bratu–Gelfand problem for radial operators. Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Journal of Differential Equations 184, 283–298 (2002). https://doi.org/10.1006/jdeq.2001.4151 [49] T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- T.A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 167–195 (2004). 10.1145/992200.992205 [50] T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) T.A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30(2), 196–199 (2004). 10.1145/992200.992206 [51] V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V.T. Zhukov, N. Novikova, O.B. Feodoritova, On direct solving conjugate heat transfer of gas mixture and solid body. KIAM Preprint 2023(12) (2023). https://doi.org/10.20948/prepr-2023-12 [52] V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. 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- V. Grimm, Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012). 10.1007/s10543-011-0367-8 [53] M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Exponential Euler and backward Euler methods for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Lobachevskii Journal of Mathematics 44(1), 10–19 (2023). https://doi.org/10.1134/S1995080223010067 [54] M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) M.A. Botchev, V.T. Zhukov, Adaptive iterative explicit time integration for nonlinear heat conduction problems. Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
- Lobachevskii Journal of Mathematics 45(1), 12–20 (2024). https://doi.org/10.1134/S1995080224010086 [55] D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) D. Hipp, M. Hochbruck, A. Ostermann, An exponential integrator for non-autonomous parabolic problems. ETNA 41, 497–511 (2014). http://etna.mcs.kent.edu/volumes/2011-2020/vol41/ [56] E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016) E. Hansen, A. Ostermann, High-order splitting schemes for semilinear evolution equations. BIT Numerical Mathematics 56(4), 1303–1316 (2016)
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- BIT Numerical Mathematics 56(4), 1303–1316 (2016)