Abstract:We prove stability and convergence of the Godunov scheme for a special class of genuinely nonlinear $2 \times 2$ systems of conservation laws. The class of systems, which was identified and studied by Temple, is a subset of the class of systems for which the shock wave curves and rarefaction wave curves coincide. None of the equations of gas dynamics fall into this class, but equations of this type do arise, for example, in the study of multicomponent chromatography. To our knowledge this is the first time that a numerical method other than the random choice method of Glimm has been shown to be stable in the variation norm for a coupled system of nonlinear conservation laws. This implies that subsequences converge to weak solutions of the Cauchy problem, although convergence for $2 \times 2$ systems has been proved by DiPerna using the more abstract methods of compensated compactness.
- Neal R. Amundson, Mathematical methods in chemical engineering, Prentice-Hall International Series in the Physical and Chemical Engineering Sciences, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. Volume 1: Matrices and their application. MR 0371183
- 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. MR 430536, DOI 10.1512/iumj.1977.26.26029
- Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI 10.1090/S0025-5718-1980-0551288-3
- James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
- Amiram Harten, Peter D. Lax, and Bram van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35–61. MR 693713, DOI 10.1137/1025002
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- 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. MR 0393870
- Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI 10.1002/cpa.3160130205
- Blake Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), no. 2, 781–795. MR 716850, DOI 10.1090/S0002-9947-1983-0716850-2
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI 10.1007/BF00251724 H. Rhee, R. Aris and N. R. Amundson, On the theory of multicomponent chromatography, Philos. Trans. Roy. Soc. London Ser. A 267 (1970), 4199-455.
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 288 (1985), 115-123
- MSC: Primary 35L65; Secondary 65M10
- DOI: https://doi.org/10.1090/S0002-9947-1985-0773050-X
- MathSciNet review: 773050