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We prove in this paper the weak consistency of a general finite volumeconvection operator acting on discrete functions which are possibly notpiecewise-constant over the cells of the mesh and over the time steps. Ityields an extension of the Lax-Wendroff if-theorem for general colocated ornon-colocated schemes. This result is obtained for general polygonal orpolyhedral meshes, under assumptions which, for usual practical cases,essentially boil down to a flux-consistency constraint; this latter is, up toour knowledge, novel and compares the discrete flux at a face to the mean valueover the adjacent cell of the continuous flux function applied to the discreteunknown function. We then apply this result to prove the consistency of afinite volume discretisation of a convection operator featuring a (convected)scalar variable and a (convecting) velocity field, with a staggeredapproximation, i.e. with a cell-centred approximation of the scalar variableand a face-centred approximation of the velocity.

Weak Consistency of Finite Volume Schemes for Systems of Non Linear Conservation Laws: Extension to Staggered Schemes