Convergence of an Engquist-Osher scheme for a multi-dimensional triangular system of conservation laws

Coclite, G.; Mishra, S.; Risebro, N.

doi:10.1090/S0025-5718-09-02251-0

Convergence of an Engquist-Osher scheme for a multi-dimensional triangular system of conservation laws
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by G. M. Coclite, S. Mishra and N. H. Risebro PDF

Math. Comp. 79 (2010), 71-94 Request permission

Abstract:

We consider a multi-dimensional triangular system of conservation laws. This system arises as a model of three-phase flow in porous media and includes multi-dimensional conservation laws with discontinuous coefficients as a special case. The system is neither strictly hyperbolic nor symmetric. We propose an Engquist-Osher type scheme for this system and show that the approximate solutions generated by the scheme converge to a weak solution. Numerical examples are also presented.

References

Adimurthi, Siddhartha Mishra, and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ. 2 (2005), no. 4, 783–837. MR 2195983, DOI 10.1142/S0219891605000622
Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. First-order systems and applications. MR 2284507
R. Bürger, K. H. Karlsen, N. H. Risebro, and J. D. Towers, Well-posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math. 97 (2004), no. 1, 25–65. MR 2045458, DOI 10.1007/s00211-003-0503-8
Zhangxin Chen, Guanren Huan, and Yuanle Ma, Computational methods for multiphase flows in porous media, Computational Science & Engineering, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. MR 2217767, DOI 10.1137/1.9780898718942
G. M. Coclite, K. H. Karlsen, S. Mishra and N. H. Risebro. Convergence of the viscous approximation for a triangular system of conservation laws. To appear in Unione Mat. Ital.
Michael G. Crandall and Luc Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), no. 3, 385–390. MR 553381, DOI 10.1090/S0002-9939-1980-0553381-X
Stefan Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math. 56 (1996), no. 2, 388–419. MR 1381652, DOI 10.1137/S0036139994242425
R. Eymard, T. Gallouët, M. Ghilani, and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Numer. Anal. 18 (1998), no. 4, 563–594. MR 1681074, DOI 10.1093/imanum/18.4.563
Tore Gimse and Nils Henrik Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal. 23 (1992), no. 3, 635–648. MR 1158825, DOI 10.1137/0523032
Helge Holden and Nils Henrik Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002. MR 1912206, DOI 10.1007/978-3-642-56139-9
Lars Holden, On the strict hyperbolicity of the Buckley-Leverett equations for three-phase flow in a porous medium, SIAM J. Appl. Math. 50 (1990), no. 3, 667–682. MR 1050906, DOI 10.1137/0150039
Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
Kenneth Hvistendahl Karlsen, Siddhartha Mishra, and Nils Henrik Risebro, Convergence of finite volume schemes for triangular systems of conservation laws, Numer. Math. 111 (2009), no. 4, 559–589. MR 2471610, DOI 10.1007/s00211-008-0199-x
Kenneth H. Karlsen, Michel Rascle, and Eitan Tadmor, On the existence and compactness of a two-dimensional resonant system of conservation laws, Commun. Math. Sci. 5 (2007), no. 2, 253–265. MR 2334842
K. H. Karlsen, N. H. Risebro, and J. D. Towers, $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk. 3 (2003), 1–49. MR 2024741
Kenneth Hvistendahl Karlsen and Nils Henrik Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, M2AN Math. Model. Numer. Anal. 35 (2001), no. 2, 239–269. MR 1825698, DOI 10.1051/m2an:2001114
K. H. Karlsen and J. D. Towers, Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux, Chinese Ann. Math. Ser. B 25 (2004), no. 3, 287–318. MR 2086124, DOI 10.1142/S0252959904000299
François Murat, L’injection du cône positif de $H^{-1}$ dans $W^{-1,\,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9) 60 (1981), no. 3, 309–322 (French, with English summary). MR 633007
E. Yu. Panov. Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux. Submitted.
John D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal. 38 (2000), no. 2, 681–698. MR 1770068, DOI 10.1137/S0036142999363668
John D. Towers, A difference scheme for conservation laws with a discontinuous flux: the nonconvex case, SIAM J. Numer. Anal. 39 (2001), no. 4, 1197–1218. MR 1870839, DOI 10.1137/S0036142900374974