Avec cedram.org english version Accueil Présentation Recherche avancée Tous les articles en ligne Derniers articles Rechercher un article Table des matières de ce volume | Article précédent | Article suivant Ralf Hiptmair; Cecilia PagliantiniSplitting-Based Structure Preserving Discretizations for MagnetohydrodynamicsSMAI-Journal of computational mathematics, 4 (2018), p. 225-257, doi: 10.5802/smai-jcm.34 Article PDF Class. Math.: 76W05, 65M60, 65M08, 65M12Mots clés: Magnetohydrodynamics (MHD), discrete differential forms, Finite Element Exterior Calculus (FEEC), extrusion contraction, upwinding, extended Euler equations, Orszag-Tang vortex, rotor problem AbstractWe 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. Bibliographie[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. 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