Monotone difference approximations for scalar conservation laws
Authors:
Michael G. Crandall and Andrew Majda
Journal:
Math. Comp. 34 (1980), 1-21
MSC:
Primary 65M05
DOI:
https://doi.org/10.1090/S0025-5718-1980-0551288-3
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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PH. BENILAN, Equation d’Evolution dans un Espace de Banach Quelconque, Thesis, Université de Orsay, 1972.
- Samuel Z. Burstein, Peter D. Lax, and Gary A. Sod (eds.), Lectures on combustion theory, New York University, Courant Mathematics and Computing Laboratory, New York, 1978. Lectures given in a Seminar held during spring semester at the Courant Institute, New York University, New York, 1977. MR 522091
- Edward Conway and Joel Smoller, Clobal solutions of the Cauchy problem for quasi-linear first-order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95–105. MR 192161, DOI https://doi.org/10.1002/cpa.3160190107
- Michael G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132. MR 316925, DOI https://doi.org/10.1007/BF02764657 M. G. CRANDALL & L. TARTAR, "Some relations between non expansive and order preserving mappings." (To appear.) 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.
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523 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)
- Amiram Harten, The artificial compression method for computation of shocks and contact discontinuities. I. Single conservation laws, Comm. Pure Appl. Math. 30 (1977), no. 5, 611–638. MR 438730, DOI https://doi.org/10.1002/cpa.3160300506
- A. Harten, J. M. Hyman, and P. D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR 413526, DOI https://doi.org/10.1002/cpa.3160290305
- Gray Jennings, Discrete shocks, Comm. Pure Appl. Math. 27 (1974), 25–37. MR 338594, DOI https://doi.org/10.1002/cpa.3160270103
- Kiyofumi Kojima, On the existence of discontinuous solutions of the Cauchy problem for quasi-linear first-order equations, Proc. Japan Acad. 42 (1966), 705–709. MR 212352 S. N. KRUŽKOV, "First order quasilinear equations with several space variables," Math. USSR Sb., v. 10, 1970, pp. 217-243.
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR 0350216
- Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI https://doi.org/10.1002/cpa.3160130205
- A. Y. le Roux, A numerical conception of entropy for quasi-linear equations, Math. Comp. 31 (1977), no. 140, 848–872. MR 478651, DOI https://doi.org/10.1090/S0025-5718-1977-0478651-3
- Michael Crandall and Andrew Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), no. 3, 285–314. MR 571291, DOI https://doi.org/10.1007/BF01396704
- Andrew Majda and Stanley Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32 (1979), no. 6, 797–838. MR 539160, DOI https://doi.org/10.1002/cpa.3160320605
- Shinnosuke Ôharu and Tadayasu Takahashi, A convergence theorem of nonlinear semigroups and its application to first order quasilinear equations, J. Math. Soc. Japan 26 (1974), 124–160. MR 341216, DOI https://doi.org/10.2969/jmsj/02610124 O. A. OLEĪNIK, "Discontinuous solutions of nonlinear differential equations," Amer. Math. Soc. Transl. (2), v. 26, 1963, pp. 95-172.
- Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506–517. MR 235754, DOI https://doi.org/10.1137/0705041
- A. I. Vol′pert, Spaces ${\rm BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338 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