Highorder schemes and entropy condition for nonlinear hyperbolic systems of conservation laws
Author:
J.P. Vila
Journal:
Math. Comp. 50 (1988), 5373
MSC:
Primary 65M10; Secondary 35L65
MathSciNet review:
917818
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Abstract: A systematic procedure for constructing explicit, quasi secondorder approximations to strictly hyperbolic systems of conservation laws is presented. These new schemes are obtained by correcting firstorder schemes. We prove that limit solutions satisfy the entropy inequality. In the scalar case, we prove convergence to the unique entropysatisfying solution if the initial scheme is Total Variation Decreasing (i.e., TVD) and consistent with the entropy condition. Finally, we slightly modify Harten's highorder schemes such that they obey the previous conditions and thus converge towards the "entropy" solution.
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 [3]
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 [4]
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 [5]
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 [6]
 A. Harten, J. M. Hyman & P. D. Lax, "On finite difference approximations and entropy condition for shocks," Comm. Pure. Appl. Math., v. 29, 1976, pp. 297322. MR 0413526 (54:1640)
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 A. Y. Le Roux, "Numerical stability for some equations of gas dynamics," Math. Comp., v. 37, 1981, pp. 307320. MR 628697 (82m:76044)
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 A. Y. Le Roux & P. Quesseveur, "Convergence of an antidiffusion LagrangeEuler scheme for quasilinear equations," SIAM J. Numer. Anal., v. 21, 1984, pp. 985994. MR 760627 (85m:65092)
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 E. Tadmor, "The largetime behavior of the scalar, genuinely nonlinear LaxFriedrichs scheme," Math. Comp., v. 43, 1984, pp. 353368. MR 758188 (86g:65162)
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 J. P. Vila, "Simplified Godunov schemes for systems of conservation laws," SIAM J. Numer. Anal., v. 23, 1986, pp. 11731192. MR 865949 (88d:65131)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809178181
PII:
S 00255718(1988)09178181
Article copyright:
© Copyright 1988 American Mathematical Society
