Convergence of the Finite Volume Method for Multidimensional Conservation Laws

Abstract

We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations (“flat elements” are allowed) and to Lipschitz continuous flux-functions. We treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework [Coquel and LeFloch, Math. Comp., 57 (1991), pp. 169–210 and J. Numer. Anal., 30 (1993), pp. 675–700]. From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities. These estimates are shown to be sufficient for the passage to the limit in the discrete equation. Convergence follows from DiPerna’s uniqueness result in the class of entropy measure-valued solutions.