Convergence of Upwind Finite Volume Schemes for Scalar Conservation Laws in Two Dimensions

Abstract

This paper proves the convergence of a general class of monotone finite volume methods for numerical schemes of scalar conservation laws in two dimensions on unstructured meshes. There are convergence results for fractional step methods on cartesian grids and for finite element algorithms on unstructured grids. Even for finite volume methods there are some recent results concerning the Lax–Friedrichs and the Godunov finite volume method. The proof in this paper considers a general class including the Lax–Friedrichs and the Engquist–Osher finite volume schemes, and uses a completely different idea than in previous papers to control the entropy dissipation.