DISCRETE ENTROPY AND MONOTONICITY CRITERI A FOR HYPERBOLIC CONSERVATION-LAWS

We consider the Cauchy problem for a nonlinear hyperbolic conservation law when the flux-function is strictly convex and the initial data ha s a locally finite number of extrema. We introduce a class of high ord er accurate and discrete in space and time difference schemes. By defi nition those schemes: called EMO for short, are consistent with the en tropy criterion (E) and the monotonicity property (MO). We prove the s trong convergence of these methods to the unique entropy solution. Our analysis includes tracing forward paths of extremum values, studying their limits, and the passage to the limit (in a pointwise sense) in t he traces of the approximations along those paths. Our result gives so me extension to difference approximations of the theory due to Glimm-L ax for the random choice method. We deduce that van Leer's MUSCL schem e is strongly convergent. Our result extends results by Osher, Lions-S ougadinis and Yang who treated the semi-discrete case.