Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Stability of Godunov's method for a class of $ 2\times 2$ systems of conservation laws

Authors: Randall J. LeVeque and Blake Temple
Journal: Trans. Amer. Math. Soc. 288 (1985), 115-123
MSC: Primary 35L65; Secondary 65M10
MathSciNet review: 773050
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] R. Aris and N. Amundson, Mathematical methods in chemical engineering, Vol. 2, Prentice-Hall, Englewood Cliffs, N. J., 1966. MR 0371183 (51:7404)
  • [2] K. N. Chueh, C. C. Conley and J. A. Smoller, Positively invariant regions for systems of nonlinear reaction diffusion equations, Indiana Univ. Math. J. 26 (1977), 373-392. MR 0430536 (55:3541)
  • [3] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 1-21. MR 551288 (81b:65079)
  • [4] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. MR 0194770 (33:2976)
  • [5] A. Harten, P. D. Lax, and B. van Leer, On upstream differencing and Godunov-rype schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), 35-67. MR 693713 (85h:65188)
  • [6] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 19 (1957), 537-566. MR 0093653 (20:176)
  • [7] -, Shock waves and entropy, Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603-634. MR 0393870 (52:14677)
  • [8] P. D. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217-237. MR 0120774 (22:11523)
  • [9] B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), 781-795. MR 716850 (84m:35080)
  • [10] Ronald DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1982), 27-70. MR 684413 (84k:35091)
  • [11] 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.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L65, 65M10

Retrieve articles in all journals with MSC: 35L65, 65M10

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society