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Nonlinear projection methods for multi-entropies Navier–Stokes systems

Authors: Christophe Berthon and Frédéric Coquel
Journal: Math. Comp. 76 (2007), 1163-1194
MSC (2000): Primary 65M99, 65M12; Secondary 76N15
Published electronically: February 7, 2007
MathSciNet review: 2299770
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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

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.

Keywords: Navier–Stokes equations, entropy inequalities, nonconservative products, travelling wave solutions, Godunov-type methods, discrete entropy inequalities, nonlinear projection, turbulence models
Received by editor(s): March 2, 2005
Received by editor(s) in revised form: April 16, 2006
Published electronically: February 7, 2007
Article copyright: © Copyright 2007 American Mathematical Society