On Entropy Consistency of Large Time Step Schemes I. The Godunov and Glimm Schemes

Abstract

This paper shows for scalar conservation laws with a convex flux function that the large time step approximation of weak solutions gives entropy solutions in the limit if the Courant number is between $\tfrac{1}{2}$ and 1. For flux functions with constant curvature, monotonicity, or monotone initial data the results extend to larger Courant numbers. A continuity argument gives this result also for the case of fluxes with almost constant curvature.