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.
DOI : 10.5802/smai-jcm.34
Keywords: Magnetohydrodynamics (MHD), discrete differential forms, Finite Element Exterior Calculus (FEEC), extrusion contraction, upwinding, extended Euler equations, Orszag-Tang vortex, rotor problem
Hiptmair, Ralf 1 ; Pagliantini, Cecilia 2
@article{SMAI-JCM_2018__4__225_0,
author = {Hiptmair, Ralf and Pagliantini, Cecilia},
title = {Splitting-Based {Structure} {Preserving} {Discretizations} for {Magnetohydrodynamics}},
journal = {The SMAI Journal of computational mathematics},
pages = {225--257},
year = {2018},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {4},
doi = {10.5802/smai-jcm.34},
mrnumber = {3813097},
zbl = {1416.76343},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.34/}
}
TY - JOUR AU - Hiptmair, Ralf AU - Pagliantini, Cecilia 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 - http://geodesic.mathdoc.fr/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 Hiptmair, Ralf %A Pagliantini, Cecilia %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 http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.34/ %R 10.5802/smai-jcm.34 %G en %F SMAI-JCM_2018__4__225_0
Hiptmair, Ralf; Pagliantini, Cecilia. Splitting-Based Structure Preserving Discretizations for Magnetohydrodynamics. The SMAI Journal of computational mathematics, Tome 4 (2018), pp. 225-257. doi: 10.5802/smai-jcm.34
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