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A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves

CNRS & Ecole Normale Supérieure Département de Mathématiques et Applications 45 rue d’Ulm 75230 Paris cedex 05 France; CNRS & LAMA Université Paris-Est Marne-la-Vallée 5 Boulevard Descartes Cité Descartes, Champs-sur-Marne 77454 Marne-la-Vallée cedex 2 France
Numerische Mathematik (Impact Factor: 1.33). 06/2010; 115(4):647-679. DOI: 10.1007/s00211-010-0289-4

ABSTRACT In the first part of this work Bouchut etal. (J Comput Phys 108:7–41, 2007) we introduced an approximate Riemann solver for
one-dimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete
entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation,
and derive explicit wave speed estimates satisfying the stability conditions of Bouchut etal. (J Comput Phys 108:7–41, 2007).
We present a 3-wave solver that well resolves fast waves and material contacts, and a 5-wave solver that accurately resolves
the cases when two eigenvalues coincide. A full 7-wave solver, which is highly accurate on all types of waves, will be described
in a follow-up paper. We test the solvers on one-dimensional shock tube data and smooth shear waves.

Mathematics Subject Classification (2000)76M12-65M12-76W05

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