An error estimate for finite volume methods for multidimensional conservation laws

Abstract

In this paper, an L ∞ ( L 1 ) {L^\infty }({L^1}) -error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order h 1 / 4 {h^{1/4}} , where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of continuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.