×

zbMATH — the first resource for mathematics

Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition. (English) Zbl 1380.65158

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adimurthi; Dutta, R.; Gowda, G. D. V.; Jaffré, J., Monotone \((A, B)\) entropy stable numerical scheme for scalar conservation laws with discontinuous flux, ESAIM Math. Model. Numer. Anal., 48, 6, 1725-1755, (2014) · Zbl 1308.65135
[2] Adimurthi; Jaffré, J.; Gowda, G. D. V., Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42, 1, 179-208, (2004) · Zbl 1081.65082
[3] Adimurthi; Mishra, S.; Gowda, G. D. V., Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equations, 2, 4, 783-837, (2005) · Zbl 1093.35045
[4] Adimurthi; Mishra, S.; Gowda, G. D. V., Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes, Netw. Heterog. Media, 2, 1, 127-157, (2007) · Zbl 1142.35508
[5] Adimurthi; Kumar, K. Sudarshan; Gowda, G. D. V., Second-order scheme for scalar conservation laws with discontinuous flux, Appl. Numer. Math., 80, 46-64, (2014) · Zbl 1329.65177
[6] Andreianov, B.; Cancès, C., The Godunov scheme for scalar conservation laws with discontinuous Bell-shaped flux functions, Appl. Math. Lett., 25, 11, 1844-1848, (2012) · Zbl 1253.65122
[7] Andreianov, B.; Cancès, C., Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks’ medium, Comput. Geosci., 17, 3, 551-572, (2013) · Zbl 1392.76033
[8] Andreianov, B.; Cancès, C., On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equations, 12, 2, 343-384, (2015) · Zbl 1336.35230
[9] Andreianov, B.; Karlsen, K. H.; Risebro, N. H., On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5, 3, 617-633, (2010) · Zbl 1270.35305
[10] Andreianov, B.; Karlsen, K. H.; Risebro, N. H., A theory of \(L^1\)-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Rational Mech. Anal., 201, 1, 27-86, (2011) · Zbl 1261.35088
[11] Andreianov, B.; Mitrović, D., Entropy conditions for scalar conservation laws with discontinuous flux revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32, 6, 1307-1335, (2015) · Zbl 1343.35158
[12] Audusse, E.; Pertháme, B., Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sec. A, 135, 2, 253-265, (2005) · Zbl 1071.35079
[13] Bachmann, F.; Vovelle, J., Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients, Comm. Partial Differ. Equations, 31, 1-3, 371-395, (2006) · Zbl 1102.35064
[14] Baiti, P.; Jenssen, H. K., Well-posedness for a class of \(2 \times 2\) conservation laws with \(L^\infty\)-data, J. Differ. Equations, 140, 1, 161-185, (1997) · Zbl 0892.35097
[15] Bürger, R.; García, A.; Karlsen, K. H.; Towers, J. D., On an extended clarifier-thickener model with singular source and sink terms, European J. Appl. Math., 17, 257-292, (2006) · Zbl 1201.35130
[16] Bürger, R.; García, A.; Karlsen, K. H.; Towers, J. D., A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60, 387-425, (2008) · Zbl 1200.76126
[17] Bürger, R.; Karlsen, K. H.; Klingenberg, C.; Risebro, N. H., A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units, Nonlinear Anal. Real World Appl., 4, 3, 457-481, (2003) · Zbl 1013.35052
[18] Bürger, R.; Karlsen, K. H.; Risebro, N. H.; Towers, J. D., Well-posedness in \(B V_t\) and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97, 1, 25-65, (2004) · Zbl 1053.76047
[19] Bürger, R.; Karlsen, K. H.; Towers, J. D., A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math., 65, 3, 882-940, (2005) · Zbl 1089.76061
[20] Bürger, R.; Karlsen, K. H.; Towers, J. D., An engquist-ösher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47, 3, 1684-1712, (2009) · Zbl 1201.35022
[21] Bürger, R.; Karlsen, K. H.; Torres, H.; Towers, J. D., Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units, Numer. Math., 116, 4, 579-617, (2010) · Zbl 1204.65101
[22] Chen, G.-Q.; Even, N.; Klingenberg, C., Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems, J. Differ. Equations, 245, 11, 3095-3126, (2008) · Zbl 1195.35211
[23] Crandall, M. G.; Majda, A., Monotone difference approximations for scalar conservation laws, Math. Comput., 34, 149, 1-21, (1980) · Zbl 0423.65052
[24] Crasta, G.; De Cicco, V.; De Philippis, G., Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux, Comm. Partial Differ. Equations, 40, 4, 694-726, (2015) · Zbl 1326.35197
[25] Crasta, G.; De Cicco, V.; De Philippis, G.; Ghiraldin, F., Structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness, Arch. Rational Mech. Anal., 221, 2, 961-985, (2016) · Zbl 1338.35288
[26] Diehl, S., On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26, 6, 1425-1451, (1995) · Zbl 0852.35094
[27] Diehl, S., Scalar conservation laws with discontinuous flux function. I. the viscous profile condition, Commun. Math. Phys., 176, 1, 23-44, (1996) · Zbl 0845.35067
[28] Diehl, S., A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math., 56, 2, 388-419, (1996) · Zbl 0849.35142
[29] Diehl, S., A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differ. Equations, 6, 1, 127-159, (2009) · Zbl 1180.35305
[30] Garavello, M.; Natalini, R.; Piccoli, B.; Terracina, A., Conservation laws with discontinuous flux, Netw. Heterog. Media, 2, 159-179, (2007) · Zbl 1142.35511
[31] Gimse, T.; Risebro, N. H., Riemann problems with a discontinuous flux function, Proc. 3rd Int. Conf. Hyperbolic Problems, 488-502, (1991), Studentlitteratur, Uppsala · Zbl 0789.35102
[32] Gimse, T.; Risebro, N. H., Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23, 3, 635-648, (1992) · Zbl 0776.35034
[33] Holden, H.; Risebro, N. H., Front Tracking for Hyperbolic Conservation Laws, 152, (2002), Springer-Verlag, New York · Zbl 1006.35002
[34] Jimenez, J., Mathematical analysis of a scalar multidimensional conservation law with discontinuous flux, J. Evol. Equations, 11, 3, 553-576, (2011) · Zbl 1232.35094
[35] Karlsen, K. H.; Risebro, N. H., Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, M2AN Math. Model. Numer. Anal., 35, 2, 239-269, (2001) · Zbl 1032.76048
[36] Karlsen, K. H.; Risebro, N. H.; Towers, J. D., On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient, Electron. J. Differ. Equations, 93, 23, (2002) · Zbl 1015.35049
[37] Karlsen, K. H.; Risebro, N. H.; Towers, J. D., Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient, IMA J. Numer. Anal., 22, 4, 623-664, (2002) · Zbl 1014.65073
[38] Karlsen, K. H.; Risebro, N. H.; Towers, J. D., \(L^1\)-stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., 3, 1-49, (2003) · Zbl 1036.35104
[39] Klingenberg, C.; Risebro, N. H., Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior, Comm. Partial Differ. Equations, 20, 11-12, 1959-1990, (1995) · Zbl 0836.35090
[40] LeVeque, R. J., Numerical Methods for Conservation Laws, (1992), Birkhäuser-Verlag, Basel · Zbl 0847.65053
[41] Mishra, S., Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function, SIAM J. Numer. Anal., 43, 2, 559-577, (1995) · Zbl 1096.35085
[42] Mitrovic, D., New entropy conditions for scalar conservation laws with discontinuous flux, Discrete Contin. Dynam. Syst., 30, 4, 1191-1210, (2011) · Zbl 1228.35144
[43] Panov, E. Y., On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux, J. Hyperbolic Differ. Equations, 6, 3, 525-548, (2009) · Zbl 1181.35145
[44] Risebro, N. H., Applied Wave Mathematics, An introduction to the theory of scalar conservation laws with spatially discontinuous flux functions, 395-464, (2009), Springer-Verlag, Berlin · Zbl 1191.74032
[45] Seguin, N.; Vovelle, J., Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Math. Models Meth. Appl. Sci., 13, 2, 221-257, (2003) · Zbl 1078.35011
[46] Towers, J. D., Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38, 2, 681-698, (2000) · Zbl 0972.65060
[47] Towers, J. D., A difference scheme for conservation laws with a discontinuous flux: the nonconvex case, SIAM J. Numer. Anal., 39, 4, 1197-1218, (2001) · Zbl 1055.65104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.