On the order of accuracy of the divergence theorem(Green-Gauss) method for calculating the gradient infinite volume methods


A. Syrakos, S. Varchanis, Y. Dimakopoulos, A. Goulas, J. Tsamopoulos


The divergence theorem (or Green-Gauss) gradient scheme is among the most popular methodsfor discretising the gradient operator in second-order accurate finite volume methods, with a longhistory of successful application on structured grids. This together with the ease of applicationof the scheme on unstructured grids has led to its widespread use in unstructured finite volumemethods (FVMs). However, the present study shows both theoretically and through numericaltests that the common variant of this scheme is zeroth-order accurate (it does not converge to theexact gradient) on grids of arbitrary skewness, such as typically produced by unstructured gridgeneration algorithms. Moreover, we use the scheme in the FVM solution of a diffusion (Poisson)equation problem, with both an in-house code and the popular open-source solver OpenFOAM,and observe that the zeroth-order accuracy of the gradient operator is inherited by the FVM solveras a whole. However, a simple iterative procedure that exploits the outer iterations of the FVMsolver is shown to effect first-order accuracy to the gradient and second-order accuracy to the FVMat almost no extra cost compared to the original scheme. Second-order accurate results are alsoobtained if a least-squares gradient operator is used instead.


Finite volume method; Green-Gauss gradient; divergence theorem gradient; diffusionequation; OpenFOAM