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Semidiscrete Entropy Satisfying Approximate Riemann Solvers and Application to the Suliciu Relaxation Approximation

In: Hyperbolic Problems: Theory, Numerics, Applications

Author

Listed:
  • T. Morales

    (École Normale Supérieure – DMA)

  • F. Bouchut

    (École Normale Supérieure, Département de Mathématiques et Applications)

Abstract

We establish conditions for an approximate simple Riemann solver to satisfy a semidiscrete entropy inequality. Classically, a discrete entropy inequality allows to analyze the stability of a numerical scheme for a conservative system. A semidiscrete entropy inequality gives a simpler and less restrictive approach than a fully discrete entropy inequality and leads to the definition of less restrictive conditions for numerical schemes to satisfy. First, conditions are established in an abstract framework for simple Riemann solvers to satisfy a semidiscrete entropy inequality and then the results are applied, as a particular case, to the Suliciu system. The Suliciu relaxation system is attached to the resolution of the isentropic gas dynamics system and can also handle full gas dynamics. It allows to define a simple approximate Riemann solver for gas dynamics. Conditions have already been established for the scheme to be entropy satisfying. Our approach allows to relax the conditions established in the fully discrete case and leads to the definition of a numerical scheme for gas dynamics that satisfies a semidiscrete entropy inequality while allowing exact resolution of shocks.

Suggested Citation

  • T. Morales & F. Bouchut, 2008. "Semidiscrete Entropy Satisfying Approximate Riemann Solvers and Application to the Suliciu Relaxation Approximation," Springer Books, in: Sylvie Benzoni-Gavage & Denis Serre (ed.), Hyperbolic Problems: Theory, Numerics, Applications, pages 739-746, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-75712-2_75
    DOI: 10.1007/978-3-540-75712-2_75
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