On the weak consistency of finite volumes schemes for conservation laws on general meshes

The aim of this paper is to develop some tools in order to obtain the weak consistency of (in other words, an analogue of the Lax–Wendroff theorem for) finite volume schemes for balance laws in the multi-dimensional case and under minimal regularity assumptions for the mesh. As in the seminal Lax–Wendroff paper, our approach relies on a discrete integration by parts of the weak formulation of the scheme. Doing so, a discrete gradient of the test function appears; the central argument for the scheme consistency is to remark that this discrete gradient is convergent in \(L^\infty \) weak \(\star \).