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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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Convergence of difference schemes with high resolution for conservation laws
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by Gui-Qiang Chen and Jian-Guo Liu PDF
Math. Comp. 66 (1997), 1027-1053 Request permission

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
  • MR Author ID: 249262
  • ORCID: 0000-0001-5146-3839
  • Email: gqchen@math.nwu.edu
  • Jian-Guo Liu
  • Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
  • MR Author ID: 233036
  • ORCID: 0000-0002-9911-4045
  • Email: jliu@math.temple.edu
  • Received by editor(s): April 1, 1996
  • © Copyright 1997 American Mathematical Society
  • Journal: Math. Comp. 66 (1997), 1027-1053
  • MSC (1991): Primary 65M12; Secondary 35L65
  • DOI: https://doi.org/10.1090/S0025-5718-97-00859-4
  • MathSciNet review: 1422786