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A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics (2404.09311v1)

Published 14 Apr 2024 in math.NA and cs.NA

Abstract: We present a novel high-order nodal artificial viscosity approach designed for solving Magnetohydrodynamics (MHD) equations. Unlike conventional methods, our approach eliminates the need for ad hoc parameters. The viscosity is mesh-dependent, yet explicit definition of the mesh size is unnecessary. Our method employs a multimesh strategy: the viscosity coefficient is constructed from a linear polynomial space constructed on the fine mesh, corresponding to the nodal values of the finite element approximation space. The residual of MHD is utilized to introduce high-order viscosity in a localized fashion near shocks and discontinuities. This approach is designed to precisely capture and resolve shocks. Then, high-order Runge-Kutta methods are employed to discretize the temporal domain. Through a comprehensive set of challenging test problems, we validate the robustness and high-order accuracy of our proposed approach for solving MHD equations.

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References (37)
  1. Numerical simulation of unsteady MHD flows and applications. Magnetohydrodynamics c/c of Magnitnaia Gidrodinamika, 45(2):225–232, 2009.
  2. Non-oscillatory central schemes for one- and two-dimensional MHD equations. I. J. Comput. Phys., 201(1):261–285, 2004. ISSN 0021-9991. doi: 10.1016/j.jcp.2004.05.020. URL https://doi.org/10.1016/j.jcp.2004.05.020.
  3. Dinshaw S. Balsara. Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows. J. Comput. Phys., 229(6):1970–1993, 2010. ISSN 0021-9991. doi: 10.1016/j.jcp.2009.11.018. URL https://doi.org/10.1016/j.jcp.2009.11.018.
  4. A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys., 149(2):270–292, 1999. ISSN 0021-9991. doi: 10.1006/jcph.1998.6153. URL https://doi.org/10.1006/jcph.1998.6153.
  5. Multidimensional HLLC Riemann solver for unstructured meshes—with application to Euler and MHD flows. J. Comput. Phys., 261:172–208, 2014. ISSN 0021-9991. doi: 10.1016/j.jcp.2013.12.029. URL https://doi.org/10.1016/j.jcp.2013.12.029.
  6. An FCT finite element scheme for ideal MHD equations in 1D and 2D. J. Comput. Phys., 338:585–605, 2017. ISSN 0021-9991. doi: 10.1016/j.jcp.2017.02.051. URL https://doi.org/10.1016/j.jcp.2017.02.051.
  7. A multiwave approximate Riemann solver for ideal MHD based on relaxation. I. Theoretical framework. Numer. Math., 108(1):7–42, 2007. ISSN 0029-599X. doi: 10.1007/s00211-007-0108-8. URL https://doi.org/10.1007/s00211-007-0108-8.
  8. The effect of nonzero ∇⋅𝐁⋅∇𝐁\nabla\cdot{\bf B}∇ ⋅ bold_B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys., 35(3):426–430, 1980. ISSN 0021-9991. doi: 10.1016/0021-9991(80)90079-0. URL https://doi.org/10.1016/0021-9991(80)90079-0.
  9. M. Brio and C. C. Wu. An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys., 75(2):400–422, 1988. ISSN 0021-9991. doi: 10.1016/0021-9991(88)90120-9. URL https://doi.org/10.1016/0021-9991(88)90120-9.
  10. Positivity-preserving DG and central DG methods for ideal MHD equations. J. Comput. Phys., 238:255–280, 2013. ISSN 0021-9991. doi: 10.1016/j.jcp.2012.12.019. URL https://doi.org/10.1016/j.jcp.2012.12.019.
  11. A simple finite difference scheme for multidimensional magnetohydrodynamical equations. J. Comput. Phys., 142(2):331–369, 1998. ISSN 0021-9991. doi: 10.1006/jcph.1998.5944. URL https://doi.org/10.1006/jcph.1998.5944.
  12. Monolithic parabolic regularization of the MHD equations and entropy principles. Comput. Methods Appl. Mech. Eng., 398:115269, 2022a. ISSN 0045-7825. doi: 10.1016/j.cma.2022.115269. URL https://doi.org/10.1016/j.cma.2022.115269.
  13. A high-order residual-based viscosity finite element method for the ideal MHD equations. J. Sci. Comput., 92(3):Paper No. 77, 24, 2022b. ISSN 0885-7474,1573-7691. doi: 10.1007/s10915-022-01918-4. URL https://doi.org/10.1007/s10915-022-01918-4.
  14. Structure preserving numerical methods for the ideal compressible MHD system, 2023.
  15. A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. J. Comput. Phys., 319:163–199, 2016. ISSN 0021-9991. doi: 10.1016/j.jcp.2016.05.002. URL https://doi.org/10.1016/j.jcp.2016.05.002.
  16. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004. ISBN 0-387-20574-8. doi: 10.1007/978-1-4757-4355-5. URL https://doi.org/10.1007/978-1-4757-4355-5.
  17. A maximum-principle preserving C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT finite element method for scalar conservation equations. Comput. Methods Appl. Mech. Engrg., 272:198–213, 2014. ISSN 0045-7825. doi: 10.1016/j.cma.2013.12.015. URL http://dx.doi.org/10.1016/j.cma.2013.12.015.
  18. Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C. R. Math. Acad. Sci. Paris, 346(13-14):801–806, 2008. ISSN 1631-073X. doi: 10.1016/j.crma.2008.05.013.
  19. Implementation of the entropy viscosity method. Technical Report 4015, KTH, Numerical Analysis, NA, 2011a. QC 20110720.
  20. Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys., 230(11):4248–4267, 2011b.
  21. Finite element-based invariant-domain preserving approximation of hyperbolic systems: Beyond second-order accuracy in space. Computer Methods in Applied Mechanics and Engineering, 418:116470, 2024. ISSN 0045-7825. doi: https://doi.org/10.1016/j.cma.2023.116470. URL https://www.sciencedirect.com/science/article/pii/S0045782523005947.
  22. Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations. J. Comput. Phys., 407:109230, 18, 2020. ISSN 0021-9991. doi: 10.1016/j.jcp.2020.109230. URL https://doi.org/10.1016/j.jcp.2020.109230.
  23. Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comput., 22/23:413–442, 2005. ISSN 0885-7474. doi: 10.1007/s10915-004-4146-4. URL https://doi.org/10.1007/s10915-004-4146-4.
  24. Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field. J. Comput. Phys., 230(12):4828–4847, 2011. ISSN 0021-9991. doi: 10.1016/j.jcp.2011.03.006. URL https://doi.org/10.1016/j.jcp.2011.03.006.
  25. A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations. J. Comput. Phys., 410:109390, 28, 2020. ISSN 0021-9991. doi: 10.1016/j.jcp.2020.109390. URL https://doi.org/10.1016/j.jcp.2020.109390.
  26. Murtazo Nazarov. Convergence of a residual based artificial viscosity finite element method. Comput. Math. Appl., 65(4):616–626, 2013. ISSN 0898-1221. doi: 10.1016/j.camwa.2012.11.003. URL http://dx.doi.org/10.1016/j.camwa.2012.11.003.
  27. Residual-based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods. Internat. J. Numer. Methods Fluids, 71(3):339–357, 2013. ISSN 0271-2091. doi: 10.1002/fld.3663. URL http://dx.doi.org/10.1002/fld.3663.
  28. Numerical investigation of a viscous regularization of the Euler equations by entropy viscosity. Comput. Methods Appl. Mech. Engrg., 317:128–152, 2017. ISSN 0045-7825.
  29. VMS finite element for MHD and reduced-MHD in Tokamak plasmas. Research Report RR-8892, Inria Sophia Antipolis ; Université de Nice-Sophia Antipolis, March 2016. URL https://inria.hal.science/hal-01294788.
  30. Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech., 90(1):129–143, 1979. ISSN 0022-1120. doi: 10.1017/S002211207900210X. URL https://doi.org/10.1006/10.1017/S002211207900210X.
  31. A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys., 154(2):284–309, 1999. ISSN 0021-9991. doi: 10.1006/jcph.1999.6299. URL https://doi.org/10.1006/jcph.1999.6299.
  32. Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods. J. Comput. Phys., 229(20):7649–7671, 2010. ISSN 0021-9991,1090-2716. doi: 10.1016/j.jcp.2010.06.018. URL https://doi.org/10.1016/j.jcp.2010.06.018.
  33. Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton-Krylov-AMG. Comput. Methods Appl. Mech. Engrg., 304:1–25, 2016. ISSN 0045-7825. doi: 10.1016/j.cma.2016.01.019. URL https://doi.org/10.1016/j.cma.2016.01.019.
  34. Athena: a new code for astrophysical MHD. Astrophys. J., Suppl. Ser., 178(1):137, 2008.
  35. M. Torrilhon. Exact solver and uniqueness conditions for riemann problems of ideal magnetohydrodynamics. Technical report, Zurich: Seminar for Applied Mathematics, ETH, 2002.
  36. A discontinuous Galerkin method for the viscous MHD equations. J. Comput. Phys., 152(2):608–641, 1999. ISSN 0021-9991. doi: 10.1006/jcph.1999.6248. URL https://doi.org/10.1006/jcph.1999.6248.
  37. A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics. SIAM J. Sci. Comput., 40(5):B1302–B1329, 2018. ISSN 1064-8275. doi: 10.1137/18M1168042. URL https://doi.org/10.1137/18M1168042.

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