zbMATH — the first resource for mathematics

Invariant regions for systems of conservation laws. (English) Zbl 0535.35056
Necessary and sufficient conditions are given for a region in \({\mathbb{R}}^ n\) to be invariant for (Glimm) solutions of the system of n conservation laws \(u_ t+f(u)_ x=0.\) A number of examples are given, and some observation are made about the invariance of such regions for certain finite difference approximations of solutions of systems of conservation laws.

35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
35L40 First-order hyperbolic systems
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
Full Text: DOI
[1] Rutherford Aris and Neal R. Amundson, Mathematical methods in chemical engineering, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Volume 2: First-order partial differential equations with applications; Prentice-Hall International Series in the Physical and Chemical Engineering Sciences.
[2] K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), no. 2, 373 – 392. · Zbl 0368.35040 · doi:10.1512/iumj.1977.26.26029 · doi.org
[3] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697 – 715. · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 · doi.org
[4] David Hoff, A finite difference scheme for a system of two conservation laws with artificial viscosity, Math. Comp. 33 (1979), no. 148, 1171 – 1193. · Zbl 0447.65056
[5] David Hoff and Joel Smoller, Error bounds for finite-difference approximations for a class of nonlinear parabolic systems, Math. Comp. 45 (1985), no. 171, 35 – 49. · Zbl 0613.65096
[6] J. A. Smoller and J. L. Johnson, Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 32 (1969), 169 – 189. · Zbl 0167.10204 · doi:10.1007/BF00247508 · doi.org
[7] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537 – 566. · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 · doi.org
[8] Peter Lax, Shock waves and entropy, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603 – 634. · Zbl 0268.35014
[9] Takaaki Nishida and Joel Smoller, A class of convergent finite difference schemes for certain nonlinear parabolic systems, Comm. Pure Appl. Math. 36 (1983), no. 6, 785 – 808. · Zbl 0535.65063 · doi:10.1002/cpa.3160360605 · doi.org
[10] Stanley Osher, Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 179 – 204. · Zbl 0457.76035
[11] Stanley Osher and Fred Solomon, Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comp. 38 (1982), no. 158, 339 – 374. · Zbl 0483.65055
[12] Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. · Zbl 0129.37203
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.