Convergence analysis of a cell centered finite volume diffusion operator on non-orthogonal polyhedral meshes

TL;DR

This paper provides a convergence analysis of a finite volume diffusion operator on non-orthogonal polyhedral meshes, demonstrating its robustness and accuracy, and introduces a least squares gradient approach for improved convergence.

Contribution

It offers the first convergence proof for this widely used finite volume method and interprets the correction as an anisotropic operator, enhancing understanding and implementation.

Findings

Convergence is achieved under weak mesh distortion assumptions.

The corrected operator is equivalent to an anisotropic diffusion operator.

A least squares gradient approach can attain second order accuracy.

Abstract

A simple but successful strategy for building a discrete diffusion operator in finite volume schemes of industrial use is to correct the standard two-point flux approximation with a term accounting for the local mesh non-orthogonality. Practical experience with a variety of different mesh typologies, including non-orthogonal tetrahedral, hexahedral and polyhedral meshes, has shown that this discrete diffusion operator is accurate and robust whenever the mesh is not too distorted and sufficiently regular. In this work, we show that this approach can be interpreted as equivalent to introducing an anisotropic operator that accounts for the preferential directions induced by the local mesh non-orthogonality. This allows to derive a convergence analysis of the corrected method under a quite weak global assumption on mesh distortion. This convergence proof, which is obtained for the first time for this finite volume method widely employed in industrial software packages such as OpenFOAM, provides a reference framework on how to interpret some of its variants commonly implemented in commercial finite volume codes. Numerical experiments are presented that confirm the accuracy and robustness of the results. Furthermore, we also show empirically that a least square approach to the gradient computation can provide second order convergence even when the mild non-orthogonality condition on the mesh is violated.