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Equilibrium schemes for scalar conservation laws with stiff sources

Authors: Ramaz Botchorishvili, Benoit Perthame and Alexis Vasseur
Journal: Math. Comp. 72 (2003), 131-157
MSC (2000): Primary 65M06, 65M12, 35L65
Published electronically: November 20, 2001
MathSciNet review: 1933816
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Abstract: We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no $BV$ estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

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Additional Information

Ramaz Botchorishvili
Affiliation: VIAM, Tbilissi State University, 2 University Street, 380043 Tbilissi, Georgia

Benoit Perthame
Affiliation: INRIA, M3N, domaine de Voluceau, BP 105, F78153 Le Chesnay
Address at time of publication: ENS, DMA, 45, rue d’Ulm, F75230 Paris cédex 05, France

Alexis Vasseur
Affiliation: Laboratoire J.A. Dieudonné, UMR 6621, Université Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 02, France

Keywords: Hyperbolic conservation laws, kinetic formulation, stiff source terms, upwind schemes, convergence
Received by editor(s): March 29, 2000
Received by editor(s) in revised form: January 3, 2001
Published electronically: November 20, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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