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

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Abstract

In the first part of this work Bouchut et al. (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 et al. (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.

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Authors and Affiliations

  1. Département de Mathématiques et Applications, CNRS & Ecole Normale Supérieure, 45 rue d’Ulm, 75230, Paris cedex 05, France

    François Bouchut

  2. CNRS & LAMA Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes Cité Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée cedex 2, France

    François Bouchut

  3. Department of Mathematics, Würzburg University, Am Hubland, 97074, Würzburg, Germany

    Christian Klingenberg

  4. High Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO, 80307-3000, USA

    Knut Waagan

Authors
  1. François Bouchut
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  2. Christian Klingenberg
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  3. Knut Waagan
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Correspondence to François Bouchut.

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Bouchut, F., Klingenberg, C. & Waagan, K. A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves. Numer. Math. 115, 647–679 (2010). https://doi.org/10.1007/s00211-010-0289-4

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  • DOI: https://doi.org/10.1007/s00211-010-0289-4

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