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Entropy-satisfying relaxation method with large time-steps for Euler IBVPs

Authors: Frédéric Coquel, Quang Long Nguyen, Marie Postel and Quang Huy Tran
Journal: Math. Comp. 79 (2010), 1493-1533
MSC (2010): Primary 65M08; Secondary 35L04
Published electronically: February 23, 2010
MathSciNet review: 2630001
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Abstract: This paper could have been given the title: ``How to positively and implicitly solve Euler equations using only linear scalar advections.'' The new relaxation method we propose is able to solve Euler-like systems--as well as initial and boundary value problems--with real state laws at very low cost, using a hybrid explicit-implicit time integration associated with the Arbitrary Lagrangian-Eulerian formalism. Furthermore, it possesses many attractive properties, such as: (i) the preservation of positivity for densities; (ii) the guarantee of min-max principle for mass fractions; (iii) the satisfaction of entropy inequality, under an expressible bound on the CFL ratio. The main feature that will be emphasized is the design of this optimal time-step, which takes into account data not only from the inner domain but also from the boundary conditions.

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Additional Information

Frédéric Coquel
Affiliation: UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Quang Long Nguyen
Affiliation: CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Marie Postel
Affiliation: Département Mathématiques Appliquées, Institut Français du Pétrole, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France

Quang Huy Tran
Affiliation: Département Mathématiques Appliquées, Institut Français du Pétrole, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France

Keywords: Euler equations, multiphase flow, initial boundary value problems, explicit-implicit, relaxation methods, Lagrange-projection, entropy-satisfying, positivity-preserving
Received by editor(s): December 31, 2007
Received by editor(s) in revised form: February 27, 2009
Published electronically: February 23, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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