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

  • Authors:

  • François Bouchut;Christian Klingenberg;Knut Waagan

  • Affiliations:

  • CNRS & Ecole Normale Supéérieure, Département de Mathématiques et Applications and CNRS & LAMA Universitéé Paris-Est Marne-la-Vallée, France;Würzburg University, Department of Mathematics, Am Hubland, 97074, Würzburg, Germany;National Center for Atmospheric Research, High Altitude Observatory, P.O. Box 3000, 80307-3000, Boulder, CO, USA

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2010

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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.