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Mathematics of Computation

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Monotone difference approximations for scalar conservation laws

Authors: Michael G. Crandall and Andrew Majda
Journal: Math. Comp. 34 (1980), 1-21
MSC: Primary 65M05
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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Keywords: Conservation laws, shock waves difference approximations, entropy conditions
Article copyright: © Copyright 1980 American Mathematical Society