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Multiderivative time integration methods preserving nonlinear functionals via relaxation (2311.03883v2)

Published 7 Nov 2023 in math.NA and cs.NA

Abstract: We combine the recent relaxation approach with multiderivative Runge-Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.

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References (70)
  1. “Implementation of second derivative general linear methods” In Calcolo 57.20 Springer, 2020 DOI: 10.1007/s10092-020-00370-w
  2. “Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes” In The SMAI Journal of Computational Mathematics 8, 2022, pp. 125–160 DOI: 10.5802/smai-jcm.82
  3. “Error propagation in the numerical integration of solitary waves. The regularized long wave equation” In Applied Numerical Mathematics 36.2-3 Elsevier, 2001, pp. 197–217 DOI: 10.1016/S0168-9274(99)00148-8
  4. Arpit Babbar, Sudarshan Kumar Kenettinkara and Praveen Chandrashekar “Lax-Wendroff flux reconstruction method for hyperbolic conservation laws” In Journal of Computational Physics 467.111423 Elsevier, 2022 DOI: 10.1016/j.jcp.2022.111423
  5. Thomas Brooke Benjamin, Jerry Lloyd Bona and John Joseph Mahony “Model equations for long waves in nonlinear dispersive systems” In Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 272.1220 The Royal Society London, 1972, pp. 47–78 DOI: 10.1098/rsta.1972.0032
  6. “Julia: A Fresh Approach to Numerical Computing” In SIAM Review 59.1 SIAM, 2017, pp. 65–98 DOI: 10.1137/141000671
  7. Abhijit Biswas and David I Ketcheson “Accurate Solution of the Nonlinear Schrödinger Equation via Conservative Multiple-Relaxation ImEx Methods”, 2023 arXiv:2309.02324 [math.NA]
  8. Abhijit Biswas and David I Ketcheson “Multiple relaxation Runge-Kutta methods for conservative dynamical systems” In Journal of Scientific Computing 97.4, 2023 DOI: 10.1007/s10915-023-02312-4
  9. “On the Preservation of Invariants by Explicit Runge-Kutta Methods” In SIAM Journal on Scientific Computing 28.3 SIAM, 2006, pp. 868–885 DOI: 10.1137/04061979X
  10. “Error growth in the numerical integration of periodic orbits” In Mathematics and Computers in Simulation 81.12 Elsevier, 2011, pp. 2646–2661 DOI: 10.1016/j.matcom.2011.05.007
  11. “Projection methods preserving Lyapunov functions” In BIT Numerical Mathematics 50.2 Springer, 2010, pp. 223–241 DOI: 10.1007/s10543-010-0259-3
  12. B Cano and Jesus Maria Sanz-Serna “Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems” In SIAM Journal on Numerical Analysis 34.4 SIAM, 1997, pp. 1391–1417 DOI: 10.1137/S0036142995281152
  13. Robert PK Chan and Angela YJ Tsai “On explicit two-derivative Runge-Kutta methods” In Numerical Algorithms 53 Springer, 2010, pp. 171–194 DOI: 10.1007/s11075-009-9349-1
  14. J De Frutos and Jesus Maria Sanz-Serna “Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation” In Numerische Mathematik 75.4 Springer, 1997, pp. 421–445 DOI: 10.1007/s002110050247
  15. Angel Durán and Jesus Maria Sanz-Serna “The numerical integration of relative equilibrium solutions. Geometric theory” In Nonlinearity 11.6 IOP Publishing, 1998 DOI: 10.1088/0951-7715/11/6/008
  16. Angel Durán and Jesus Maria Sanz-Serna “The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation” In IMA Journal of Numerical Analysis 20.2 Oxford University Press, 2000, pp. 235–261 DOI: 10.1093/imanum/20.2.235
  17. Travis C Fisher and Mark H Carpenter “High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains” In Journal of Computational Physics 252 Elsevier, 2013, pp. 518–557 DOI: 10.1016/j.jcp.2013.06.014
  18. Matteo Frigo and Steven G Johnson “The design and implementation of FFTW3” In Proceedings of the IEEE 93.2 IEEE, 2005, pp. 216–231 DOI: 10.1109/JPROC.2004.840301
  19. “High order entropy preserving ADER-DG scheme” In Applied Mathematics and Computation 440.127644 Elsevier, 2023 DOI: 10.1016/j.amc.2022.127644
  20. Gregor Josef Gassner, Andrew Ross Winters and David A Kopriva “Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations” In Journal of Computational Physics 327 Elsevier, 2016, pp. 39–66 DOI: 10.1016/j.jcp.2016.09.013
  21. “High Order Strong Stability Preserving Multiderivative Implicit and IMEX Runge-Kutta Methods with Asymptotic Preserving Properties” In SIAM Journal on Numerical Analysis 60.1 SIAM, 2022, pp. 423–449 DOI: 10.1137/21M1403175
  22. Sigal Gottlieb, David I Ketcheson and Chi-Wang Shu “Strong stability preserving Runge-Kutta and multistep time discretizations” Singapore: World Scientific, 2011
  23. “Multistep-multistage-multiderivative methods for ordinary differential equations” In Computing (Arch. Elektron. Rechnen) 11.3, 1973, pp. 287–303 DOI: 10.1007/BF02252917
  24. Ernst Hairer, Christian Lubich and Gerhard Wanner “Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations” 31, Springer Series in Computational Mathematics Berlin Heidelberg: Springer-Verlag, 2006 DOI: 10.1007/3-540-30666-8
  25. Ernst Hairer, Syvert Paul Nørsett and Gerhard Wanner “Solving Ordinary Differential Equations I: Nonstiff Problems” 8, Springer Series in Computational Mathematics Berlin Heidelberg: Springer-Verlag, 2008 DOI: 10.1007/978-3-540-78862-1
  26. Alexander Jaust, Jochen Schütz and David C Seal “Implicit multistage two-derivative discontinuous Galerkin schemes for viscous conservation laws” In Journal of Scientific Computing 69 Springer, 2016, pp. 866–891 DOI: 10.1007/s10915-016-0221-x
  27. Shinhoo Kang and Emil M Constantinescu “Entropy-Preserving and Entropy-Stable Relaxation IMEX and Multirate Time-Stepping Methods” In Journal of Scientific Computing 93, 2022, pp. 23 DOI: 10.1007/s10915-022-01982-w
  28. “On Turan type implicit Runge-Kutta methods” In Computing 9, 1972, pp. 317–325 DOI: 10.1007/BF02241605
  29. David I Ketcheson “Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms” In SIAM Journal on Numerical Analysis 57.6 Society for IndustrialApplied Mathematics, 2019, pp. 2850–2870 DOI: 10.1137/19M1263662
  30. David I Ketcheson and Hendrik Ranocha “Computing with B-series” In ACM Transactions on Mathematical Software, 2022 DOI: 10.1145/3573384
  31. “Systems of conservation laws” In Communications on Pure and Applied Mathematics 13.2 Wiley Online Library, 1960, pp. 217–237 DOI: 10.1002/cpa.3160130205
  32. “A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations” In ESAIM: Mathematical Modelling and Numerical Analysis 55.6 EDP Sciences, 2021, pp. 2567–2608 DOI: 10.1051/m2an/2021065
  33. Dongfang Li, Xiaoxi Li and Zhimin Zhang “Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs” In Mathematics of Computation American Mathematical Society, 2022 DOI: 10.1090/mcom/3766
  34. Dongfang Li, Xiaoxi Li and Zhimin Zhang “Linearly implicit and high-order energy-preserving relaxation schemes for highly oscillatory Hamiltonian systems” In Journal of Computational Physics Elsevier, 2023 DOI: 10.1016/j.jcp.2023.111925
  35. Viktor Linders, Hendrik Ranocha and Philipp Birken “Resolving Entropy Growth from Iterative Methods” In BIT Numerical Mathematics, 2023 DOI: 10.1007/s10543-023-00992-w
  36. MathWorks “MATLAB”, 2022 URL: https:/mathworks.com/products/matlab.html
  37. “A conservative fully-discrete numerical method for the regularized shallow water wave equations” In SIAM Journal on Scientific Computing 42, 2021 DOI: 10.1137/20M1364606
  38. A. Moradi, A. Abdi and J. Farzi “Strong stability preserving second derivative general linear methods with Runge-Kutta stability” In Journal of Scientific Computing 85.1, 2020, pp. Paper No. 1, 39 DOI: 10.1007/s10915-020-01306-w
  39. A. Quarteroni, R. Sacco and F. Saleri “Numerical Mathematics” Springer, 2007 DOI: 10.1007/b98885
  40. Hendrik Ranocha “Comparison of Some Entropy Conservative Numerical Fluxes for the Euler Equations” In Journal of Scientific Computing 76.1 Springer, 2018, pp. 216–242 DOI: 10.1007/s10915-017-0618-1
  41. Hendrik Ranocha “Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators” In Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 134, Lecture Notes in Computational Science and Engineering Cham: Springer, 2020, pp. 525–535 DOI: 10.1007/978-3-030-39647-3_42
  42. Hendrik Ranocha “On Strong Stability of Explicit Runge-Kutta Methods for Nonlinear Semibounded Operators” In IMA Journal of Numerical Analysis 41.1 Oxford University Press, 2021, pp. 654–682 DOI: 10.1093/imanum/drz070
  43. Hendrik Ranocha “SummationByPartsOperators.jl: A Julia library of provably stable semidiscretization techniques with mimetic properties” In Journal of Open Source Software 6.64 The Open Journal, 2021, pp. 3454 DOI: 10.21105/joss.03454
  44. Hendrik Ranocha, Lisandro Dalcin and Matteo Parsani “Fully-Discrete Explicit Locally Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations” In Computers and Mathematics with Applications 80.5 Elsevier, 2020, pp. 1343–1359 DOI: 10.1016/j.camwa.2020.06.016
  45. Hendrik Ranocha and Gregor J Gassner “Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes” In Communications on Applied Mathematics and Computation, 2021 DOI: 10.1007/s42967-021-00148-z
  46. “Stability of step size control based on a posteriori error estimates”, 2023 DOI: 10.48550/arXiv.2307.12677
  47. Hendrik Ranocha and David I Ketcheson “Energy Stability of Explicit Runge-Kutta Methods for Nonautonomous or Nonlinear Problems” In SIAM Journal on Numerical Analysis 58.6 Society for IndustrialApplied Mathematics, 2020, pp. 3382–3405 DOI: 10.1137/19M1290346
  48. Hendrik Ranocha and David I Ketcheson “Relaxation Runge-Kutta Methods for Hamiltonian Problems” In Journal of Scientific Computing 84.1 Springer Nature, 2020 DOI: 10.1007/s10915-020-01277-y
  49. Hendrik Ranocha, Lajos Lóczi and David I Ketcheson “General Relaxation Methods for Initial-Value Problems with Application to Multistep Schemes” In Numerische Mathematik 146 Springer Nature, 2020, pp. 875–906 DOI: 10.1007/s00211-020-01158-4
  50. Hendrik Ranocha, Manuel Luna and David I Ketcheson “On the Rate of Error Growth in Time for Numerical Solutions of Nonlinear Dispersive Wave Equations” In Partial Differential Equations and Applications 2.6, 2021, pp. 76 DOI: 10.1007/s42985-021-00126-3
  51. Hendrik Ranocha, Dimitrios Mitsotakis and David I Ketcheson “A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations” In Communications in Computational Physics 29.4 Global Science Press, 2021, pp. 979–1029 DOI: 10.4208/cicp.OA-2020-0119
  52. “Relaxation Runge-Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations” In SIAM Journal on Scientific Computing 42.2 Society for IndustrialApplied Mathematics, 2020, pp. A612–A638 DOI: 10.1137/19M1263480
  53. “Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws” In ACM Transactions on Mathematical Software, 2023 DOI: 10.1145/3625559
  54. “Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing” In Proceedings of the JuliaCon Conferences 1.1 The Open Journal, 2022, pp. 77 DOI: 10.21105/jcon.00077
  55. “Reproducibility repository for "Multiderivative time integration methods preserving nonlinear functionals via relaxation"”, 2023_multiderivative_relaxation, 2023 DOI: 10.5281/zenodo.10057727
  56. Hendrik Ranocha, Jochen Schütz and Eleni Theodosiou “Functional-preserving predictor-corrector multiderivative schemes” In Proceedings in Applied Mathematics and Mechanics, 2023 DOI: 10.1002/pamm.202300025
  57. J. Revels, M. Lubin and T. Papamarkou “Forward-Mode Automatic Differentiation in Julia”, 2016 DOI: 10.48550/arXiv.1607.07892
  58. Jesus Maria Sanz-Serna “An explicit finite-difference scheme with exact conservation properties” In Journal of Computational Physics 47.2 Elsevier, 1982, pp. 199–210 DOI: 10.1016/0021-9991(82)90074-2
  59. “A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics” In Journal of Computational Physics 442 Elsevier, 2021, pp. 110467 DOI: 10.1016/j.jcp.2021.110467
  60. Jochen Schütz and David C Seal “An asymptotic preserving semi-implicit multiderivative solver” In Applied Numerical Mathematics 160 Elsevier, 2021, pp. 84–101 DOI: 10.1016/j.apnum.2020.09.004
  61. Jochen Schütz, David C Seal and Alexander Jaust “Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations” In Journal of Scientific Computing 73 Springer, 2017, pp. 1145–1163 DOI: 10.1007/s10915-017-0485-9
  62. Jochen Schütz, David C Seal and Jonas Zeifang “Parallel-in-time high-order multiderivative IMEX solvers” In Journal of Scientific Computing 90.1 Springer, 2022, pp. 54 DOI: 10.1007/s10915-021-01733-3
  63. David C Seal, Yaman Güçlü and Andrew J Christlieb “High-order multiderivative time integrators for hyperbolic conservation laws” In Journal of Scientific Computing 60 Springer, 2014, pp. 101–140 DOI: 10.1007/s10915-013-9787-8
  64. Angela YJ Tsai, Robert PK Chan and Shixiao Wang “Two-derivative Runge-Kutta methods for PDEs using a novel discretization approach” In Numerical Algorithms 65 Springer, 2014, pp. 687–703 DOI: 10.1007/s11075-014-9823-2
  65. “Derivation of three-derivative Runge-Kutta methods” In Numerical Algorithms 74.1, 2017, pp. 247–265 DOI: 10.1007/s11075-016-0147-2
  66. P Turán “On the theory of the mechanical quadrature” In Acta Universitatis Szegediensis Acta Scientiarum Mathematicarum 12, 1950, pp. 30–37
  67. “Entropy stable discontinuous Galerkin methods for balance laws in non-conservative form: Applications to the Euler equations with gravity” In Journal of Computational Physics 468 Elsevier, 2022, pp. 111507 DOI: 10.1016/j.jcp.2022.111507
  68. Jonas Zeifang, Jochen Schütz and David C Seal “Stability of implicit multiderivative deferred correction methods” In BIT Numerical Mathematics Springer, 2022, pp. 1–17 DOI: 10.1007/s10543-022-00919-x
  69. Jonas Zeifang, Arjun Thenery Manikantan and Jochen Schütz “Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method” In Applied Mathematics and Computation 457.128198, 2023 DOI: 10.1016/j.amc.2023.128198
  70. “Highly efficient invariant-conserving explicit Runge-Kutta schemes for nonlinear Hamiltonian differential equations” In Journal of Computational Physics Elsevier, 2020, pp. 109598 DOI: 10.1016/j.jcp.2020.109598
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