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On the convergence of difference approximations to scalar conservation laws. (English) Zbl 0637.65091
A unified treatment of two-level, explicit in time, second-order resolution (SOR), total-variation diminishing (TVD) finite difference approximations to scalar conservation laws is presented. After introducing a modified flux and a viscosity coefficient the authors set up a viscosity form of these schemes, which are assumed only to be both of conservation and incremental form. Considering both three point stencil and wider stencil schemes the authors show a recipe for converting three point TVD schemes to five point SOR-TVD schemes. While discussing the existence of a cell entropy inequality they show that such an inequality for all entropies implies the scheme to be an E scheme. By enforcing a single discrete entropy inequality the authors also prove convergence for TVD-SOR schemes approximating convex or concave conservation laws.
Reviewer: V.Kamen

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Software:
SHASTA
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