Splitting-Based Structure Preserving Discretizations for Magnetohydrodynamics
The SMAI Journal of computational mathematics, Volume 4 (2018), pp. 225-257.

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.

Published online:
DOI: 10.5802/smai-jcm.34
Classification: 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

Ralf Hiptmair 1; Cecilia Pagliantini 2

1 Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland
2 MCSS, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{SMAI-JCM_2018__4__225_0,
     author = {Ralf Hiptmair and Cecilia Pagliantini},
     title = {Splitting-Based {Structure} {Preserving} {Discretizations} for {Magnetohydrodynamics}},
     journal = {The SMAI Journal of computational mathematics},
     pages = {225--257},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {4},
     year = {2018},
     doi = {10.5802/smai-jcm.34},
     zbl = {1416.76343},
     mrnumber = {3813097},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.34/}
}
TY  - JOUR
AU  - Ralf Hiptmair
AU  - Cecilia Pagliantini
TI  - Splitting-Based Structure Preserving Discretizations for Magnetohydrodynamics
JO  - The SMAI Journal of computational mathematics
PY  - 2018
SP  - 225
EP  - 257
VL  - 4
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.34/
DO  - 10.5802/smai-jcm.34
LA  - en
ID  - SMAI-JCM_2018__4__225_0
ER  - 
%0 Journal Article
%A Ralf Hiptmair
%A Cecilia Pagliantini
%T Splitting-Based Structure Preserving Discretizations for Magnetohydrodynamics
%J The SMAI Journal of computational mathematics
%D 2018
%P 225-257
%V 4
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.34/
%R 10.5802/smai-jcm.34
%G en
%F SMAI-JCM_2018__4__225_0
Ralf Hiptmair; Cecilia Pagliantini. Splitting-Based Structure Preserving Discretizations for Magnetohydrodynamics. The SMAI Journal of computational mathematics, Volume 4 (2018), pp. 225-257. doi : 10.5802/smai-jcm.34. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.34/

[1] D. N. Arnold; G. Awanou Finite element differential forms on cubical meshes, Math. Comp., Volume 83 (2014) no. 288, pp. 1551-1570 | DOI | MR | Zbl

[2] D. N. Arnold; D. Boffi; F. Bonizzoni Finite element differential forms on curvilinear cubic meshes and their approximation properties, Numer. Math., Volume 129 (2015) no. 1, pp. 1-20 | DOI | MR | Zbl

[3] D. N. Arnold; R. S. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., Volume 15 (2006), pp. 1-155 | DOI | MR | Zbl

[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.), Volume 47 (2010) no. 2, pp. 281-354 | DOI | MR | Zbl

[5] D. S. Balsara Total variation diminishing scheme for adiabatic and isothermal magnetohydrodynamics, The Astrophysical Journal Supplement Series, Volume 116 (1998) no. 1, pp. 133-153 | DOI

[6] D. S. Balsara Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, The Astrophysical Journal Supplement Series, Volume 151 (2004) no. 1, pp. 149-184 | DOI

[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., Volume 149 (1999) no. 2, pp. 270-292 | DOI | MR | Zbl

[8] A. Bossavit Extrusion, contraction: their discretization via Whitney forms, COMPEL, Volume 22 (2003) no. 3, pp. 470-480 | DOI | MR | Zbl

[9] M. Brio; C.-C. Wu An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys., Volume 75 (1988) no. 2, pp. 400-422 | DOI | MR | Zbl

[10] P. G. Ciarlet The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978 | Zbl

[11] B. Einfeldt On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., Volume 25 (1988) no. 2, pp. 294-318 | DOI | MR | Zbl

[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., Volume 43 (2009) no. 5, pp. 825-852 | DOI | Numdam | MR | Zbl

[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., Volume 9 (2011) no. 2, pp. 324-362 | DOI | MR | Zbl

[14] Franz G. Fuchs; S. Mishra; N. H. Risebro Splitting based finite volume schemes for ideal MHD equations, J. Comput. Phys., Volume 228 (2009) no. 3, pp. 641-660 | DOI | MR | Zbl

[15] H. Goedbloed; S. Poedts Principles of Magnetohydrodynamics, Cambridge University Press, 2004 | DOI

[16] S. Gottlieb; C.-W. Shu; E. Tadmor Strong stability-preserving high-order time discretization methods, SIAM Rev., Volume 43 (2001) no. 1, pp. 89-112 | DOI | MR | Zbl

[17] J.-L. Guermond; R. Pasquetti Entropy-based nonlinear viscosity for Fourier approximations of conservation laws, C. R. Math. Acad. Sci. Paris, Volume 346 (2008) no. 13-14, pp. 801-806 | DOI | MR | Zbl

[18] J.-L. Guermond; R. Pasquetti; B. Popov Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., Volume 230 (2011) no. 11, pp. 4248-4267 | DOI | MR

[19] A. Harten; B. Engquist; S. Osher; S. R. Chakravarthy Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys., Volume 71 (1987) no. 2, pp. 231-303 | DOI | MR | Zbl

[20] A. Harten; P. D. Lax; B. van Leer On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., Volume 25 (1983) no. 1, pp. 35-61 | DOI | MR | Zbl

[21] H. Heumann; R. Hiptmair Extrusion contraction upwind schemes for convection-diffusion problems (2008) (SAM Report 2008-30, Seminar for Applied Mathematics, ETH Zürich)

[22] H. Heumann; R. Hiptmair Convergence of lowest order semi-Lagrangian schemes, Found. Comput. Math., Volume 13 (2013) no. 2, pp. 187-220 | DOI | MR | Zbl

[23] H. Heumann; R. Hiptmair; C. Pagliantini Stabilized Galerkin for transient advection of differential forms, Discrete Contin. Dyn. Syst. Ser. S, Volume 9 (2016) no. 1, pp. 185-214 | MR | Zbl

[24] R. Hiptmair Canonical construction of finite elements, Math. Comp., Volume 68 (1999) no. 228, pp. 1325-1346 | DOI | MR | Zbl

[25] R. Hiptmair Finite elements in computational electromagnetism, Acta Numer., Volume 11 (2002), pp. 237-339 | DOI | MR | Zbl

[26] K. Hu; Y. Ma; J. Xu Stable finite element methods preserving ·B=0 exactly for MHD models, Numer. Math., Volume 135 (2017) no. 2, pp. 371-396 | DOI | MR | Zbl

[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, Volume 195 (2011) no. 20, pp. 1-16

[28] K. H. Karlsen; S. Mishra; N. H. Risebro Semi-Godunov schemes for multiphase flows in porous media, Appl. Numer. Math., Volume 59 (2009) no. 9, pp. 2322-2336 | DOI | MR | Zbl

[29] T. J. Linde A three-dimensional adaptive multifluid MHD model of the heliosphere, University of Michigan, Ann Arbor, MI (1998) (Ph. D. Thesis)

[30] T. Miyoshi; K. Kusano A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., Volume 208 (2005) no. 1, pp. 315-344 | DOI | MR | Zbl

[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., Volume 11 (2011) no. 2, pp. 131-149 | DOI | MR | Zbl

[32] S. A. Orszag; C.-M. Tang Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., Volume 90 (1979) no. 1, pp. 129-143 | DOI

[33] C. Pagliantini Computational Magnetohydrodynamics with Discrete Differential Forms, Dis. no 23781, ETH Zürich (2016) (Ph. D. Thesis)

[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 | Zbl

[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., Volume 96 (2004) no. 4, pp. 771-800 | DOI | MR | Zbl

[37] C.-W. Shu; S. Osher Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys., Volume 83 (1989) no. 1, pp. 32-78 | DOI | MR | Zbl

[38] M. Tabata A finite element approximation corresponding to the upwind finite differencing, Mem. Numer. Math., Volume 4 (1977), pp. 47-63 | MR | Zbl

[39] E. F. Toro; M. Spruce; W. Speares Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, Volume 4 (1994) no. 1, pp. 25-34 | DOI | Zbl

[40] G. Tóth The ·B=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., Volume 161 (2000) no. 2, pp. 605-652 | DOI | MR | Zbl

[41] K. Yee Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Transactions on Antennas and Propagation, Volume 14 (1966) no. 3, p. 302-30 | DOI | Zbl

Cited by Sources: