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

T Gallouët (1) , R Herbin (1) , J.-C Latché (2)

1 I2M - Institut de Mathématiques de Marseille
2 IRSN/PSN-RES/SA2I - Service des Agressions Internes et des risques Industriels

We prove in this paper the weak 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 if-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 then apply this result 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.

Mots clés: 2010 AMS Subject Classification. Primary 65M08; 76N15; Secondary 65M12; 76N19

Domaines: Analyse numérique [math.NA]

T Gallouët, R Herbin, J.-C Latché. Lax–Wendroff consistency of finite volume schemes for systems of non linear conservation laws: extension to staggered schemes. SeMa Journal, 2022, ⟨10.1007/s40324-021-00263-0⟩. ⟨hal-03168277v2⟩