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Monotone difference approximations for scalar conservation laws

Authors: Michael G. Crandall and Andrew Majda
Journal: Math. Comp. 34 (1980), 1-21
MSC: Primary 65M05
MathSciNet review: 551288
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Abstract: A complete self-contained treatment of the stability and convergence properties of conservation-form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme.

The results are general enough to include, for instance, Godunov's scheme, the upwind scheme (differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations.

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  • [1] PH. BENILAN, Equation d'Evolution dans un Espace de Banach Quelconque, Thesis, Université de Orsay, 1972.
  • [2] S. BURSTEIN, P. D. LAX & G. SOD, Lectures on Combustion Theory, Courant Mathematics and Computing Laboratory Report, September 1978. MR 522091 (80b:76001)
  • [3] E. CONWAY & J. SMOLLER, "Global solutions of the Cauchy problem for quasilinear first order equations in several space variables," Comm. Pure Appl. Math., v. 19, 1966, pp. 95-105. MR 0192161 (33:388)
  • [4] M. G. CRANDALL, "The semigroup approach to first order quasilinear equations in several space variables," Israel J. Math., v. 12, 1972, pp. 108-132. MR 0316925 (47:5473)
  • [5] M. G. CRANDALL & L. TARTAR, "Some relations between non expansive and order preserving mappings." (To appear.)
  • [6] A. DOUGLIS, Lectures on Discontinuous Solutions of First Order Nonlinear Partial Differential Equations in Several Space Variables, North British Symposium on Partial Differential Equations, 1972.
  • [7] N. DUNFORD & J. T. SCHWARTZ, Linear Operators, Part 1 : General Theory, Pure and Appl. Math., Vol. 7, Interscience, New York, London, 1958. MR 0117523 (22:8302)
  • [8] S. K. GODUNOV, "Finite difference methods for numerical computations of discontinuous solution of equations of fluid dynamics," Mat. Sb., v. 47, 1959, pp. 271-295. (Russian)
  • [9] A. HARTEN, "The artificial compression method for computation of shocks and contact discontinuities: I. Single conservation laws," Comm. Pure Appl. Math., v. 39, 1977, pp. 611-638. MR 0438730 (55:11637)
  • [10] A. HARTEN, J. M. HYMAN & P. D. LAX, "On finite difference approximations and entropy conditions for shocks," Comm. Pure Appl. Math., v. 29, 1976, pp. 297-322. MR 0413526 (54:1640)
  • [11] G. JENNINGS, "Discrete shocks," Comm. Pure Appl. Math., v. 27, 1974, pp. 25-37. MR 0338594 (49:3358)
  • [12] K. KOJIMA, "On the existence of discontinuous solutions of the Cauchy problem for quasilinear first order equations," Proc. Japan Acad., v. 42, 1966, pp. 705-709. MR 0212352 (35:3225)
  • [13] S. N. KRUŽKOV, "First order quasilinear equations with several space variables," Math. USSR Sb., v. 10, 1970, pp. 217-243.
  • [14] P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Series in Applied Mathematics #11. MR 0350216 (50:2709)
  • [15] P. D. LAX & B. WENDROFF, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217-237. MR 0120774 (22:11523)
  • [16] A. Y. LE ROUX, "A numerical conception of entropy for quasi-linear equations," Math. Comp., v. 31, 1977, pp. 848-872. MR 0478651 (57:18128)
  • [17] A. MAJDA & M. CRANDALL, Numer. Math. (To appear.) MR 571291 (81j:65101)
  • [18] A. MAJDA & S. OSHER, "Numerical viscosity and the entropy condition," Comm. Pure Appl. Math. (To appear.) MR 539160 (80j:65031)
  • [19] S. OHARU & T. TAKAHASHI, "A convergence theorem of nonlinear semigroups and its application to first order quasilinear equations," J. Math. Soc. Japan, v. 26, 1974, pp. 124-160. MR 0341216 (49:5966)
  • [20] O. A. OLEĪNIK, "Discontinuous solutions of nonlinear differential equations," Amer. Math. Soc. Transl. (2), v. 26, 1963, pp. 95-172.
  • [21] G. STRANG, "On the construction and comparison of difference schemes," SIAM J. Numer. Anal., v. 15, 1968, pp. 506-517. MR 0235754 (38:4057)
  • [22] A. I. VOL'PERT, "The spaces BV and quasilinear equations," Math. USSR Sb., v. 2, 1967, pp. 225-267. MR 0216338 (35:7172)
  • [23] N. N. KUZNECOV & S. A. VOLOŠIN, "On monotone difference approximations for a first-order quasi-linear equation," Soviet Math. Dokl., v. 17, 1976, pp. 1203-1206.

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Keywords: Conservation laws, shock waves difference approximations, entropy conditions
Article copyright: © Copyright 1980 American Mathematical Society

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