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

TL;DR

This paper extends the weak consistency proof of finite volume schemes for nonlinear conservation laws to include staggered schemes on complex meshes, broadening the theoretical foundation for numerical methods in fluid dynamics.

Contribution

It introduces a novel flux-consistency constraint and proves weak consistency for staggered finite volume schemes on polygonal meshes, extending classical theorems.

Findings

Proves weak consistency for general finite volume convection operators.

Extends Lax-Wendroff theorem to colocated and non-colocated schemes.

Establishes flux-consistency as a key assumption for practical schemes.

Abstract

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.