On Wavewise Entropy Inequality for High Resolution Schemes II: Fully Discrete MUSCL Schemes with Exact Evolution in Small Time

Abstract

In this paper, we extend the framework and the convergence criteria of wavewise entropy inequality of [H. Yang, Math. Comp., 65 (1996), pp. 45--67] to fully discrete high resolution schemes satisfying certain TVD nonoscillatory conditions. For the Cauchy problem of convex conservation laws in one space dimension, we use one of the criteria to prove the convergence of the MUSCL scheme toward the entropy solution, assuming that each time step of the scheme consists of a minmod slope limiter, an exact time evolution, and a standard cell averaging; the CFL number is less than 0.5; and the initial condition is of bounded variation.