Convergence of Vanishing Viscosity Approximations of 2 x 2 Triangular Systems of Multi-Dimensional Conservation Laws
Coclite, G. M. ; Karlsen, K. H. ; Mishra, S. ; Risebro, N. H.
Bollettino dell'Unione Matematica Italiana, Tome 2 (2009), p. 275-284 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

We consider a multidimensional triangular system of conservation laws. These equations arise in models of three phase flows in porous media and include multi dimensional conservation laws with discontinuous coefficients as special cases. We study approximate solutions of these equations constructed by the vanishing viscosity method and show that the approximate solutions converge to a weak solution of the multi-dimensional triangular system.

Publié le : 2009-02-01
@article{BUMI_2009_9_2_1_275_0,
     author = {G. M. Coclite and K. H. Karlsen and S. Mishra and N. H. Risebro},
     title = {Convergence of Vanishing Viscosity Approximations of 2 x 2 Triangular Systems of Multi-Dimensional Conservation Laws},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {2},
     year = {2009},
     pages = {275-284},
     zbl = {1178.35246},
     mrnumber = {2493656},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_1_275_0}
}

Coclite, G. M.; Karlsen, K. H.; Mishra, S.; Risebro, N. H. Convergence of Vanishing Viscosity Approximations of 2 x 2 Triangular Systems of Multi-Dimensional Conservation Laws. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 275-284. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_1_275_0/

[1] Mishra, Adimurthi-Siddhartha - Veerappa Gowda, G. D., Optimal entropy solutions for conservation laws with discontinuous flux. Journal of Hyp. Diff. Eqns., 2 (4) (2005), 783-837. | MR 2195983 | Zbl 1093.35045

[2] Benzoni-Gavage, S. - Serre, D., Multidimensional hyperbolic partial differential equations. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2007. First-order systems and applications. | MR 2284507

[3] Bürger, R. - Karlsen, K. H. - Risebro, N. H. - Towers, J. D., Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units. Numer. Math., 97 (1) (2004), 25-65. | MR 2045458 | Zbl 1053.76047

[4] Chen, Z., Computational methods for multiphase flows in porous media. Computational science and Engineering, SIAM, 2006. | MR 2217767 | Zbl 1092.76001

[5] Coclite, G. M. - Mishra, S. - Risebro, N. H., Convergence of Engquist-Osher scheme for a multidimensional triangular system of conservation laws, submitted. | Zbl 1369.65108

[6] Diehl, S., A conservation law with point source and discontinuous flux function modeling continuous sedimentation. SIAM J. Appl. Math., 56 (2) (1995), 1980-2007. | MR 1381652

[7] Gimse, T. - Risebro, N. H., Solution of Cauchy problem for a conservation law with discontinuous flux function. SIAM J. Math. Anal, 23 (3) (1992), 635-648. | MR 1158825 | Zbl 0776.35034

[8] Hörmander, L., Lectures on non-linear hyperbolic differential equations. Mathematics and Applications, Vol 26, Springer, 1997. | MR 1466700

[9] Karlsen, K. H. - Mishra, S. - Risebro, N. H., Convergence of finite volume schemes for triangular systems of conservation laws. Submitted. | MR 2471610 | Zbl 1190.65133

[10] Karlsen, K. H. - Rascle, M. - Tadmor, E., On the existence and compactness of a two-dimensional resonant system of conservation laws. Commun. Math. Sci., 5 (2), 2007. | MR 2334842 | Zbl 1165.35412

[11] Karlsen, K. H. - Risebro, N. H. - Towers, J. D., L1 stability for entropy solution of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk., 3 (2003), 49 pages. | MR 2024741 | Zbl 1036.35104

[12] Karlsen, K. H. - Towers, J. D., Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux. Chinese Ann. Math. Ser. B, 25 (3) (2004), 287-318. | MR 2086124 | Zbl 1112.65085

[13] Murat, F., L'injection du cône positif de H-1 dans W-1,q est compacte pour tout q<2. J. Math. Pures Appl. (9), 60 (3), (1981), 309-322. | MR 633007 | Zbl 0471.46020

[14] Panov, E. Yu., Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux. Submitted. | MR 2592291 | Zbl 1200.35052

[15] Towers, J. D., Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal., 38 (2) (2000), 681-698. | MR 1770068 | Zbl 0972.65060

[16] Towers, J. D., A difference scheme for conservation laws with a discontinuous flux, the nonconvex case. SIAM J. Numer. Anal., 39 (4) (2001), 1197-1218. | MR 1870839 | Zbl 1055.65104