On the convergence of difference approximations to scalar conservation laws

Osher, Stanley; Tadmor, Eitan

doi:10.1090/S0025-5718-1988-0917817-X

On the convergence of difference approximations to scalar conservation laws
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by Stanley Osher and Eitan Tadmor PDF

Math. Comp. 50 (1988), 19-51 Request permission

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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J. Comput. Phys.