On the convergence of difference approximations to scalar conservation laws.

*(English)*Zbl 0637.65091A 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 |

##### Keywords:

entropy inequality; viscosity form; explicit in time; second-order resolution; total-variation diminishing; finite difference approximations; scalar conservation laws; convergence##### Software:

SHASTA
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\textit{S. Osher} and \textit{E. Tadmor}, Math. Comput. 50, No. 181, 19--51 (1988; Zbl 0637.65091)

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