A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

TL;DR

This paper critically examines popular gradient discretisation methods in finite volume methods, revealing their accuracy limitations on unstructured meshes and proposing more consistent schemes for improved numerical results.

Contribution

The study provides a theoretical and numerical comparison of divergence theorem and least-squares gradient schemes, introducing a new iterative scheme for better consistency.

Findings

DT gradient is only second-order on structured meshes

LS gradient achieves second-order accuracy, unlike DT on unstructured meshes

A new iterative scheme improves the consistency of the DT gradient

Abstract

Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely believed to be second-order accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM.