On convergence of monotone finite difference schemes with variable spatial differencing
Author:
Richard Sanders
Journal:
Math. Comp. 40 (1983), 91106
MSC:
Primary 65M05; Secondary 65M10, 65M15
MathSciNet review:
679435
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Abstract: Monotone finite difference schemes used to approximate solutions of scalar conservation laws have the advantage that these approximations can be proved to converge to the proper solution as the mesh size tends to zero. The greatest disadvantage in using such approximating schemes is the computational expense encountered since monotone schemes can have at best first order accuracy. Computation savings and effective accuracy could be gained if the spatial mesh were refined in regions of expected rapid solution variation. In this paper we prove that standard monotone difference schemes, (satisfying a fairly unrestrictive CFL condition), converge to the "correct" physical solution even in the case when a nonuniform spatial mesh is employed.
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 M. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 122. MR 551288 (81b:65079)
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 N. Dunford & J. Schwartz, Linear Operators, Part 1: General Theory, Pure and Appl. Math., vol. VII, Interscience, New York, 1958.
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 B. Engquist & S. Osher, "Stable and entropy satisfying approximations for transonic flow calculations," Math. Comp., v. 34, 1980, pp. 4575. MR 551290 (81b:65082)
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 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. 297322. MR 0413526 (54:1640)
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 S. N. Kružkov, "First order quasilinear equations with several space variables," Math. USSR Sb., v. 10, 1970, pp. 217243.
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 N. N. Kuznetsov, "On stable methods for solving nonlinear first order partial differential equations in the class of discontinuous functions," Topics in Numerical Analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976), pp. 183197. MR 0657786 (58:31874)
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 P. D. Lax, "Hyperbolic systems of conservation laws and the mathematical theory of shock waves," SIAM Reg. Conf. Series in Appl. Math., v. 11, 1972. MR 0350216 (50:2709)
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 P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217237. MR 0120774 (22:11523)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198306794356
PII:
S 00255718(1983)06794356
Article copyright:
© Copyright 1983
American Mathematical Society
