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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Nonlinear projection methods for multi-entropies Navier–Stokes systems
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by Christophe Berthon and Frédéric Coquel PDF
Math. Comp. 76 (2007), 1163-1194 Request permission

Abstract:

This paper is devoted to the numerical approximation of the compressible Navier–Stokes equations with several independent entropies. Various models for complex compressible materials typically enter the proposed framework. The striking novelty over the usual Navier–Stokes equations stems from the generic impossibility of recasting equivalently the present system in full conservation form. Classical finite volume methods are shown to grossly fail in the capture of viscous shock solutions that are of primary interest in the present work. To enforce for validity a set of generalized jump conditions that we introduce, we propose a systematic and effective correction procedure, the so-called nonlinear projection method, and prove that it preserves all the stability properties satisfied by suitable Godunov-type methods. Numerical experiments assess the relevance of the method when exhibiting approximate solutions in close agreement with exact solutions.
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Additional Information
  • Christophe Berthon
  • Affiliation: MAB, UMR 5466 CNRS, Université Bordeaux I, 351 cours de la libération, 33405 Talence Cedex, France
  • Email: Christophe.Berthon@math.u-bordeaux1.fr
  • Frédéric Coquel
  • Affiliation: CNRS and Laboratoire Jacques-Louis Lions, UMR 7598, Tour 55-65, Université Pierre et Marie Curie, BC 187, 75252 Paris Cedex 05, France.
  • Email: coquel@ann.jussieu.fr
  • Received by editor(s): March 2, 2005
  • Received by editor(s) in revised form: April 16, 2006
  • Published electronically: February 7, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: Math. Comp. 76 (2007), 1163-1194
  • MSC (2000): Primary 65M99, 65M12; Secondary 76N15
  • DOI: https://doi.org/10.1090/S0025-5718-07-01948-5
  • MathSciNet review: 2299770