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An All-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes

Part of: Basic methods in fluid mechanics Incompressible inviscid fluids Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  22 June 2016

Christophe Chalons ,
Mathieu Girardin  and
Samuel Kokh
Christophe Chalons*
Affiliation:
LMV - UMR 8100, Univ. Versailles Saint-Quentin-en-Yvelines, UFR des Sciences, Bâtiment Fermat, 45 avenue des Etats-Unis, 78035 Versailles cedex, France
Mathieu Girardin*
Affiliation:
DEN/DANS/DM2S/STMF/LMEC CEA Saclay, bât. 454 PC 47, 91191 Gif sur Yvette Cedex, France LRC MANON, Laboratoire de Recherche Conventionné CEA/DEN/DANS/DM2S and UPMC-CNRS/LJLL
Samuel Kokh*
Affiliation:
Maison de la Simulation USR 3441, Digiteo Labs, bât. 565, PC 190, CEA Saclay, 91191 Gif-sur-Yvette, France DEN/DANS/DM2S/STMF, CEA Saclay, 91191 Gif-sur-Yvette, France
*
*Corresponding author. Email addresses:christophe.chalons@uvsq.fr (C. Chalons), mathieu.girardin@cea.fr (M. Girardin), samuel.kokh@cea.fr (S. Kokh)
*Corresponding author. Email addresses:christophe.chalons@uvsq.fr (C. Chalons), mathieu.girardin@cea.fr (M. Girardin), samuel.kokh@cea.fr (S. Kokh)
*Corresponding author. Email addresses:christophe.chalons@uvsq.fr (C. Chalons), mathieu.girardin@cea.fr (M. Girardin), samuel.kokh@cea.fr (S. Kokh)
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Abstract

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and efficient semi implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.

Keywords

Gas dynamics equationslow-Mach regimefinite volume schemesall-regime schemesLagrange-Projection like schemeslarge time-steps

MSC classification

Secondary: 35L65: Conservation laws 35L40: First-order hyperbolic systems 65M08: Finite volume methods 76N15: Gas dynamics, general 76M12: Finite volume methods 76B99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 188 - 233
Copyright
Copyright © Global-Science Press 2016 

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