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Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint

Published online by Cambridge University Press:  15 April 2004

Florent Berthelin
Florent Berthelin*
Affiliation:
Laboratoire J.A. Dieudonné, UMR 6621 CNRS, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France. [email protected].
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Abstract

We study in this paper some numerical schemes for hyperbolic systemswith unilateral constraint. In particular, we deal with the scalar case, the isentropicgas dynamics system and the full-gas dynamics system.We prove the convergence of the scheme to an entropy solutionof the isentropicgas dynamics with unilateral constraint on the density and mass loss.We also study the non-trivial steady states of the system.

Keywords

Numerical schemeconservation laws with constraintconvergence of schemeentropy schemegas dynamics.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 3 , May 2003 , pp. 479 - 494
Copyright
© EDP Sciences, SMAI, 2003

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References

Berthelin, F., Existence and weak stability for a two-phase model with unilateral constraint. Math. Models Methods Appl. Sci. 12 (2002) 249–272. CrossRef
Berthelin, F. and Bouchut, F., Solution with finite energy to a BGK system relaxing to isentropic gas dynamics. Ann. Fac. Sci. Toulouse Math. 9 (2000) 605–630. CrossRef
Berthelin, F. and Bouchut, F., Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics. Asymptot. Anal. 31 (2002) 153–176.
F. Berthelin and F. Bouchut, Weak solutions for a hyperbolic system with unilateral constraint and mass loss. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear).
R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Rapport INRIA RR-3891.
Bouchut, F., Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113–170. CrossRef
F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. (to appear).
Chen, G.-Q. and LeFloch, P.G., Entropy flux-splittings for hyperbolic conservation laws I, General framework. Comm. Pure Appl. Math. 48 (1995) 691–729. CrossRef
Chen, G.-Q. and LeFloch, P.G., Entropies and flux-splittings for the isentropic Euler equations. Chinese Ann. Math. Ser. B 22 (2001) 145–158. CrossRef
B. Després, Equality or convex inequality constraints and hyperbolic systems of conservation laws with entropy. Preprint (2001).
Weinan, E., Rykov, Y.G. and Sinai, Y.G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys. 177 (1996) 349–380. CrossRef
E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Mathématiques & Applications 3/4, Ellipses, Paris (1991).
Gosse, L. and Le Roux, A.-Y., A well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 543–546.
Greenberg, J.M. and Le Roux, A.-Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. CrossRef
Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source term. ESAIM: M2AN 35 (2001) 631–645. CrossRef
Kružkov, S.N., First order quasilinear equations in several independant variables. Mat. Sb. 81 (1970) 285–255; Mat. Sb 10 (1970) 217–243.
Lattanzio, C. and Serre, D., Convergence of a relaxation scheme for hyperbolic systems of conservation laws. Numer. Math. 88 (2001) 121–134. CrossRef
Lévi, L., Obstacle problems for scalar conservation laws. ESAIM: M2AN 35 (2001) 575–593. CrossRef
Lions, P.-L., Perthame, B. and Souganidis, P.E., Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49 (1996) 599–638. 3.0.CO;2-5>CrossRef
Mignot, F. and Puel, J.-P., Inéquations variationnelles et quasivariationnelles hyperboliques du premier ordre. J. Math. Pures Appl. 55 (1976) 353–378.