Convergence of an EngquistOsher scheme for a multidimensional triangular system of conservation laws
Authors:
G. M. Coclite, S. Mishra and N. H. Risebro
Journal:
Math. Comp. 79 (2010), 7194
MSC (2000):
Primary 65L06, 35L65; Secondary 76S05
Published electronically:
July 10, 2009
MathSciNet review:
2552218
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Abstract: We consider a multidimensional triangular system of conservation laws. This system arises as a model of threephase flow in porous media and includes multidimensional conservation laws with discontinuous coefficients as a special case. The system is neither strictly hyperbolic nor symmetric. We propose an EngquistOsher 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.
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 1.
 Adimurthi, S. Mishra and G. D. V. Gowda.
Optimal entropy solutions for conservation laws with discontinuous fluxfunctions. J. Hyperbolic Differ. Equ., 2 (4), 783837, 2005. MR 2195983 (2007g:35144)
 2.
 S. BenzoniGavage and D. Serre.
Multidimensional hyperbolic partial differential equations, Firstorder systems and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. MR 2284507 (2008k:35002)
 3.
 R. Bürger, K.H. Karlsen, N.H. Risebro and J.D. Towers.
Wellposedness in and convergence of a difference scheme for continuous sedimentation in ideal clarifierthickener units. Numer. Math., 97 (1):2565, 2004. MR 2045458 (2004m:35175)
 4.
 Z. Chen, G. Huan and Y. Ma.
Computational methods for multiphase flows in porous media. Computational Science and Engineering, SIAM, 2006. MR 2217767 (2007c:76070)
 5.
 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.
 6.
 M. G. Crandall and L. Tartar.
Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc 78 (3), 385  390, 1980. MR 553381 (81a:47054)
 7.
 S. Diehl.
A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math., 56 (2):19802007, 1995. MR 1381652 (97a:35145)
 8.
 R. Eymard, T. Gallouet, M. Ghilani and R. Herbin.
Error estimates for the approximate solutions of nonlinear conservation laws given by finite difference schemes. IMA J. Numer. Anal., 18(4), 563594, 1998. MR 1681074 (2000b:65180)
 9.
 T. Gimse and N. H. Risebro.
Solution of Cauchy problem for a conservation law with discontinuous flux function. SIAM J. Math. Anal, 23 (3): 635648, 1992. MR 1158825 (93e:35070)
 10.
 H. Holden and N. H. Risebro.
Front tracking for hyperbolic conservation laws, volume 152 of Applied Mathematical Sciences. SpringerVerlag, New York, 2002. MR 1912206 (2003e:35001)
 11.
 L. Holden.
On the strict hyperbolicity of the BuckleyLeverett equations for threephase flow in a porous medium. SIAM J. Appl. Math., 50 (3), 667682, 1990. MR 1050906 (91c:35098)
 12.
 L. Hörmander.
Lectures on nonlinear hyperbolic differential equations. Mathematics and Applications, Vol. 26, Springer, 1997. MR 1466700 (98e:35103)
 13.
 K. H. Karlsen, S. Mishra and N. H. Risebro.
Convergence of finite volume schemes for triangular systems of conservation laws. Numer. Math., 111(4), 2009, 559589. MR 2471610
 14.
 K. H. Karlsen, M. Rascle and E. Tadmor.
On the existence and compactness of a twodimensional resonant system of conservation laws. Commun. Math. Sci, 5(2), 2007, 253265. MR 2334842 (2008j:35121)
 15.
 K. H. Karlsen, N.H. Risebro and J.D. Towers.
stability for entropy solutions of nonlinear degenerate parabolic convectiondiffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk.,3, 2003, 49 pages. MR 2024741 (2004j:35149)
 16.
 K. H. Karlsen and N. H. Risebro.
Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. M2AN Math. Model. Numer. Anal., 35 (2), 239269, 2001. MR 1825698 (2002b:35138)
 17.
 K. H. Karlsen and J. D. Towers.
Convergence of the LaxFriedrichs scheme and stability for conservation laws with a discontinous spacetime dependent flux. Chinese Ann. Math. Ser. B, 25(3):287318, 2004. MR 2086124 (2005h:65145)
 18.
 F. Murat.
L'injection du cône positif de dans est compacte pour tout . J. Math. Pures Appl. (9), 60(3):309322, 1981. MR 633007 (83b:46045)
 19.
 E. Yu. Panov.
Existence and strong precompactness properties for entropy solutions of a firstorder quasilinear equation with discontinuous flux. Submitted.
 20.
 J.D. Towers.
Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal., 38(2):681698, 2000. MR 1770068 (2001f:65098)
 21.
 J.D. Towers.
A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal., 39(4): 11971218, 2001. MR 1870839 (2002k:65131)
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Additional Information
G. M. Coclite
Affiliation:
Department of Mathematics, University of Bari, via E. Orabona 4, I–70125 Bari, Italy
Email:
coclitegm@dm.uniba.it
S. Mishra
Affiliation:
Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway
Email:
siddharm@math.uio.no
N. H. Risebro
Affiliation:
Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway
Email:
nilshr@math.uio.no
DOI:
http://dx.doi.org/10.1090/S0025571809022510
PII:
S 00255718(09)022510
Received by editor(s):
July 30, 2008
Received by editor(s) in revised form:
December 13, 2008
Published electronically:
July 10, 2009
Additional Notes:
The authors thank Kenneth H. Karlsen for many useful discussions. This research is supported in part by the Research Council of Norway. This paper was written as part of the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–09.
Article copyright:
© Copyright 2009
American Mathematical Society
