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Convergence of difference schemes with high resolution for conservation laws

Authors: Gui-Qiang Chen and Jian-Guo Liu
Journal: Math. Comp. 66 (1997), 1027-1053
MSC (1991): Primary 65M12; Secondary 35L65
MathSciNet review: 1422786
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Abstract: We are concerned with the convergence of Lax-Wendroff type schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions-the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed $L^{p}$ estimates of the approximate solutions, $H^{-1}$ compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws.

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Additional Information

Gui-Qiang Chen
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Jian-Guo Liu
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Keywords: Conservation laws, convergence, entropy solution, Lax-Wendroff scheme
Received by editor(s): April 1, 1996
Article copyright: © Copyright 1997 American Mathematical Society