Convergence of second-order, entropy stable methods for multi-dimensional conservation laws

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern 0316 Oslo, Norway

^* Corresponding author: nelabja12@gmail.com; neelabjc@math.uio.no

Received: 12 June 2019
Accepted: 10 December 2019

Abstract

High-order accurate, entropy stable numerical methods for hyperbolic conservation laws have attracted much interest over the last decade, but only a few rigorous convergence results are available, particularly in multiple space dimensions. In this paper we show how the entropy stability of one such method, which is semi-discrete in time, yields a (weak) bound on oscillations. Under the assumption of L^∞-boundedness of the approximations we use compensated compactness to prove convergence to a weak solution satisfying at least one entropy condition.