A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework

Authors:
François Bouchut;Christian Klingenberg;Knut Waagan
Affiliations:
CNRS and Ecole Normale Supérieure, Département de Mathématiques et Applications, 45 rue d’Ulm, 75230, Paris Cedex 05, France;Würzburg University, Departement of Mathematics, Am Hubland, 97074, Würzburg, Germany;Center of Mathematics for Applications, P.O. Box 1053, Blindern, 0316, Oslo, Norway
Venue:
Numerische Mathematik
Year:
2007

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Abstract

We present a relaxation system for ideal magnetohydrodynamics (MHD) that is an extension of the Suliciu relaxation system for the Euler equations of gas dynamics. From it one can derive approximate Riemann solvers with three or seven waves, that generalize the HLLC solver for gas dynamics. Under some subcharacteristic conditions, the solvers satisfy discrete entropy inequalities, and preserve positivity of density and internal energy. The subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate Riemann solver itself. The 7-wave version of the solver is able to resolve exactly all material and Alfven isolated contact discontinuities. Practical considerations and numerical results will be provided in another paper.