A local extrapolation method for hyperbolic conservation laws: the ENO and Goodman-LeVeque underlying schemes and sufficient conditions for TVD property

Abstract

We start with linear single variable conservation laws and examine the conditions under which a local extrapolation method (LEM) with upwinding underlying scheme is total variation diminishing TVD. The results are then extended to non-linear conservation laws. For this later case, we restrict ourselves to convex flux functions f, whose derivatives are positive, that is, fœ A0 and fœ A0. We next show that the Goodman-LeVeque flux satisfies the conditions for the LEM to be applied to it. We make heavy use of the CFL conditions, the geometric properties of convex functions apart from the martingle type properties of functions which are increasing, continuous, and differentiable.