Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 3, pp. 479-494.

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

DOI : 10.1051/m2an:2003038
Classification : 35A35, 35L65, 35L85, 76N15, 76T10
Mots-clés : numerical scheme, conservation laws with constraint, convergence of scheme, entropy scheme, gas dynamics 
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     title = {Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {479--494},
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     doi = {10.1051/m2an:2003038},
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     zbl = {1028.35101},
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Berthelin, Florent. Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 3, pp. 479-494. doi : 10.1051/m2an:2003038. http://www.numdam.org/articles/10.1051/m2an:2003038/

[1] F. Berthelin, Existence and weak stability for a two-phase model with unilateral constraint. Math. Models Methods Appl. Sci. 12 (2002) 249-272. | Zbl

[2] F. Berthelin and F. Bouchut, Solution with finite energy to a BGK system relaxing to isentropic gas dynamics. Ann. Fac. Sci. Toulouse Math. 9 (2000) 605-630. | EuDML | Numdam | Zbl

[3] F. Berthelin and F. Bouchut, Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics. Asymptot. Anal. 31 (2002) 153-176. | Zbl

[4] 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). | EuDML | Numdam | MR | Zbl

[5] R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Rapport INRIA RR-3891. | Zbl

[6] F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113-170. | Zbl

[7] F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models. Numer. Math. (to appear). | MR | Zbl

[8] G.-Q. Chen and P.G. Lefloch, Entropy flux-splittings for hyperbolic conservation laws I, General framework. Comm. Pure Appl. Math. 48 (1995) 691-729. | Zbl

[9] G.-Q. Chen and P.G. Lefloch, Entropies and flux-splittings for the isentropic Euler equations. Chinese Ann. Math. Ser. B 22 (2001) 145-158. | Zbl

[10] B. Després, Equality or convex inequality constraints and hyperbolic systems of conservation laws with entropy. Preprint (2001).

[11] E. Weinan, Y.G. Rykov and Y.G. Sinai, 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. | Zbl

[12] E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Mathématiques & Applications 3/4, Ellipses, Paris (1991). | MR | Zbl

[13] L. Gosse and A.-Y. Le Roux, A well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 543-546. | Zbl

[14] J.M. Greenberg and A.-Y. Le Roux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl

[15] S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source term. ESAIM: M2AN 35 (2001) 631-645. | Numdam | Zbl

[16] S.N. Kružkov, First order quasilinear equations in several independant variables. Mat. Sb. 81 (1970) 285-255; Mat. Sb 10 (1970) 217-243. | Zbl

[17] C. Lattanzio and D. Serre, Convergence of a relaxation scheme for hyperbolic systems of conservation laws. Numer. Math. 88 (2001) 121-134. | Zbl

[18] L. Lévi, Obstacle problems for scalar conservation laws. ESAIM: M2AN 35 (2001) 575-593. | Numdam | Zbl

[19] P.-L. Lions, B. Perthame and P.E. Souganidis, 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. | Zbl

[20] F. Mignot and J.-P. Puel, Inéquations variationnelles et quasivariationnelles hyperboliques du premier ordre. J. Math. Pures Appl. 55 (1976) 353-378. | Zbl

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