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Ralf Hiptmair; Cecilia Pagliantini
Splitting-Based Structure Preserving Discretizations for Magnetohydrodynamics
SMAI-Journal of computational mathematics, 4 (2018), p. 225-257, doi: 10.5802/smai-jcm.34
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Class. Math.: 76W05, 65M60, 65M08, 65M12
Keywords: Magnetohydrodynamics (MHD), discrete differential forms, Finite Element Exterior Calculus (FEEC), extrusion contraction, upwinding, extended Euler equations, Orszag-Tang vortex, rotor problem


We start from the splitting of the equations of single-fluid magnetohydrodynamics (MHD) into a magnetic induction part and a fluid part. We design novel numerical methods for the MHD system based on the coupling of Galerkin schemes for the electromagnetic fields via finite element exterior calculus (FEEC) with finite volume methods for the conservation laws of fluid mechanics. Using a vector potential based formulation, the magnetic induction problem is viewed as an instance of a generalized transient advection problem of differential forms. For the latter, we rely on an Eulerian method of lines with explicit Runge–Kutta timestepping and on structure preserving spatial upwind discretizations of the Lie derivative in the spirit of finite element exterior calculus. The balance laws for the fluid constitute a system of conservation laws with the magnetic induction field as a space and time dependent coefficient, supplied at every time step by the structure preserving discretization of the magnetic induction problem. We describe finite volume schemes based on approximate Riemann solvers adapted to accommodate the electromagnetic contributions to the momentum and energy conservation. A set of benchmark tests for the two-dimensional planar ideal MHD equations provide numerical evidence that the resulting lowest order coupled scheme has excellent conservation properties, is first order accurate for smooth solutions, conservative and stable.


[1] D. N. Arnold & G. Awanou, “Finite element differential forms on cubical meshes”, Math. Comp. 83 (2014) no. 288, p. 1551-1570
[2] D. N. Arnold, D. Boffi & F. Bonizzoni, “Finite element differential forms on curvilinear cubic meshes and their approximation properties”, Numer. Math. 129 (2015) no. 1, p. 1-20
[3] D. N. Arnold, R. S. Falk & R. Winther, “Finite element exterior calculus, homological techniques, and applications”, Acta Numer. 15 (2006), p. 1-155
[4] D. N. Arnold, R. S. Falk & R. Winther, “Finite element exterior calculus: from Hodge theory to numerical stability”, Bull. Amer. Math. Soc. (N.S.) 47 (2010) no. 2, p. 281-354
[5] D. S. Balsara, “Total variation diminishing scheme for adiabatic and isothermal magnetohydrodynamics”, The Astrophysical Journal Supplement Series 116 (1998) no. 1, p. 133-153
[6] D. S. Balsara, “Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction”, The Astrophysical Journal Supplement Series 151 (2004) no. 1, p. 149-184
[7] D. S. Balsara & D. S. Spicer, “A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations”, J. Comput. Phys. 149 (1999) no. 2, p. 270-292
[8] A. Bossavit, “Extrusion, contraction: their discretization via Whitney forms”, COMPEL 22 (2003) no. 3, p. 470-480
[9] M. Brio & C.-C. Wu, “An upwind differencing scheme for the equations of ideal magnetohydrodynamics”, J. Comput. Phys. 75 (1988) no. 2, p. 400-422
[10] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978
[11] B. Einfeldt, “On Godunov-type methods for gas dynamics”, SIAM J. Numer. Anal. 25 (1988) no. 2, p. 294-318
[12] F. G. Fuchs, K. H. Karlsen, S. Mishra & N. H. Risebro, “Stable upwind schemes for the magnetic induction equation”, M2AN Math. Model. Numer. Anal. 43 (2009) no. 5, p. 825-852
[13] F. G. Fuchs, A. D. McMurry, S. Mishra, N. H. Risebro & K. Waagan, “Approximate Riemann solvers and robust high-order finite volume schemes for multi-dimensional ideal MHD equations”, Commun. Comput. Phys. 9 (2011) no. 2, p. 324-362
[14] Franz G. Fuchs, S. Mishra & N. H. Risebro, “Splitting based finite volume schemes for ideal MHD equations”, J. Comput. Phys. 228 (2009) no. 3, p. 641-660
[15] H. Goedbloed & S. Poedts, Principles of Magnetohydrodynamics, Cambridge University Press, 2004
[16] S. Gottlieb, C.-W. Shu & E. Tadmor, “Strong stability-preserving high-order time discretization methods”, SIAM Rev. 43 (2001) no. 1, p. 89-112
[17] J.-L. Guermond & R. Pasquetti, “Entropy-based nonlinear viscosity for Fourier approximations of conservation laws”, C. R. Math. Acad. Sci. Paris 346 (2008) no. 13-14, p. 801-806
[18] J.-L. Guermond, R. Pasquetti & B. Popov, “Entropy viscosity method for nonlinear conservation laws”, J. Comput. Phys. 230 (2011) no. 11, p. 4248-4267
[19] A. Harten, B. Engquist, S. Osher & S. R. Chakravarthy, “Uniformly high-order accurate essentially nonoscillatory schemes. III”, J. Comput. Phys. 71 (1987) no. 2, p. 231-303
[20] A. Harten, P. D. Lax & B. van Leer, “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws”, SIAM Rev. 25 (1983) no. 1, p. 35-61
[21] H. Heumann & R. Hiptmair, “Extrusion contraction upwind schemes for convection-diffusion problems”, SAM Report 2008-30, Seminar for Applied Mathematics, ETH Zürich, 2008
[22] H. Heumann & R. Hiptmair, “Convergence of lowest order semi-Lagrangian schemes”, Found. Comput. Math. 13 (2013) no. 2, p. 187-220
[23] H. Heumann, R. Hiptmair & C. Pagliantini, “Stabilized Galerkin for transient advection of differential forms”, Discrete Contin. Dyn. Syst. Ser. S 9 (2016) no. 1, p. 185-214
[24] R. Hiptmair, “Canonical construction of finite elements”, Math. Comp. 68 (1999) no. 228, p. 1325-1346
[25] R. Hiptmair, “Finite elements in computational electromagnetism”, Acta Numer. 11 (2002), p. 237-339
[26] K. Hu, Y. Ma & J. Xu, “Stable finite element methods preserving $\nabla \cdot \mathbf{B}=0$ exactly for MHD models”, Numer. Math. 135 (2017) no. 2, p. 371-396
[27] R. Käppeli, S. C. Whitehouse, S. Scheidegger, U.-L. Pen & M. Liebendörfer, “FISH: a three-dimensional parallel magnetohydrodynamics code for astrophysical applications”, The Astrophysical Journal Supplement Series 195 (2011) no. 20, p. 1-16
[28] K. H. Karlsen, S. Mishra & N. H. Risebro, “Semi-Godunov schemes for multiphase flows in porous media”, Appl. Numer. Math. 59 (2009) no. 9, p. 2322-2336
[29] T. J. Linde, A three-dimensional adaptive multifluid MHD model of the heliosphere, Ph. D. Thesis, University of Michigan, Ann Arbor, MI, 1998
[30] T. Miyoshi & K. Kusano, “A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics”, J. Comput. Phys. 208 (2005) no. 1, p. 315-344
[31] P. Mullen, A. McKenzie, D. Pavlov, L. Durant, Y. Tong, E. Kanso, J. E. Marsden & M. Desbrun, “Discrete Lie advection of differential forms”, Found. Comput. Math. 11 (2011) no. 2, p. 131-149
[32] S. A. Orszag & C.-M. Tang, “Small-scale structure of two-dimensional magnetohydrodynamic turbulence”, J. Fluid Mech. 90 (1979) no. 1, p. 129-143
[33] C. Pagliantini, Computational Magnetohydrodynamics with Discrete Differential Forms, Ph. D. Thesis, Dis. no 23781, ETH Zürich, 2016
[34] P.-A. Raviart & J.-M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Springer, Berlin, 1977
[35] H.-G. Roos, M. Stynes & L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Springer-Verlag, Berlin, 2008
[36] D. Schötzau, “Mixed finite element methods for stationary incompressible magneto-hydrodynamics”, Numer. Math. 96 (2004) no. 4, p. 771-800
[37] C.-W. Shu & S. Osher, “Efficient implementation of essentially nonoscillatory shock-capturing schemes. II”, J. Comput. Phys. 83 (1989) no. 1, p. 32-78
[38] M. Tabata, “A finite element approximation corresponding to the upwind finite differencing”, Mem. Numer. Math. 4 (1977), p. 47-63
[39] E. F. Toro, M. Spruce & W. Speares, “Restoration of the contact surface in the HLL-Riemann solver”, Shock Waves 4 (1994) no. 1, p. 25-34
[40] G. Tóth, “The $\nabla \cdot B=0$ constraint in shock-capturing magnetohydrodynamics codes”, J. Comput. Phys. 161 (2000) no. 2, p. 605-652
[41] K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media”, IEEE Transactions on Antennas and Propagation 14 (1966) no. 3, p. 302-30