HIGH RESOLUTION SCHEMES AND THE ENTROPY CONDITION.

Abstract

A systematic procedure for constructing semidiscrete, second order accurate, variation diminishing, five-point band width, approximations to scalar conservation laws, is presented. These schemes are constructed to also satisfy a single discrete entropy inequality. Thus, in the convex flux case, convergence to the unique physically correct solution is proved. For hyperbolic systems of conservation laws, this construction is used to extend the first author's first order accurate scheme, and it is shown (under some minor technical hypotheses) that limit solutions satisfy an entropy inequality. Results concerning discrete shocks, a maximum principle, and maximal order of accuracy are obtained. Numerical applications are also presented.