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

The discretisation of the gradient operator is an important aspect of finite volume methods that has not received as much attention as the discretisation of other terms of partial differential equations. The most popular gradient schemes are the divergence theorem (or Green-Gauss) scheme, and the least-squares scheme. Both schemes are generally believed to be second-order accurate, but the present study shows that in fact the divergence theorem gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the least-squares gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. Furthermore, the schemes are used within a finite volume method to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the divergence theorem gradient scheme is inherited by the finite volume method as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the least-squares gradient leads to second-order accurate results. These numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM, which uses the divergence theorem gradient by default.