On the convergence of difference approximations to scalar conservation laws

Authors:
Stanley Osher and Eitan Tadmor

Journal:
Math. Comp. **50** (1988), 19-51

MSC:
Primary 65M10; Secondary 35L65, 65M05

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917817-X

MathSciNet review:
917817

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Abstract: We present a unified treatment of explicit in time, two-level, second-order resolution (SOR), total-variation diminishing (TVD), approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. We introduce a modified flux and a viscosity coefficient and obtain results in terms of the latter. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an *E* scheme on monotone (actually more general) data, hence at most only first-order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

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DOI:
https://doi.org/10.1090/S0025-5718-1988-0917817-X

Article copyright:
© Copyright 1988
American Mathematical Society