Lax–Wendroff consistency of finite volume schemes for systems of non linear conservation laws: extension to staggered schemes

We prove in this paper the Lax–Wendroff consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over the cells of the mesh and over the time steps. It yields an extension of the Lax–Wendroff theorem for general colocated or non-colocated schemes. This result is obtained for general polygonal or polyhedral meshes, under assumptions which, for usual practical cases, essentially boil down to a flux-consistency constraint; this latter is, up to our knowledge, novel and compares the discrete flux at a face to the mean value over the adjacent cell of the continuous flux function applied to the discrete unknown function. We first briefly show how this result copes with multipoint colocated schemes on general meshes. We then apply it to prove the consistency of a finite volume discretisation of a convection operator featuring a (convected) scalar variable and a (convecting) velocity field, with a staggered approximation, i.e. with a cell-centred approximation of the scalar variable and a face-centred approximation of the velocity.