A comparison of gradient approximations for use in finite-volume computational models for two-dimensional diffusion equations

Abstract

Finite-volume methods (FVMs) are now a popular choice among practitioners in scientific computation and engineering. This article focuses on generalized FVMs that can be implemented on any mesh structure. The accuracy of FVMs is primarily influenced by the numerical approximation of the flux term at the control-volume face. Here, different flux approximations are compared to identify which approximation is the most accurate, independent of the mesh structure. The accuracy of the classical two-node approximation can be improved significantly by using a local gradient reconstruction to capture the crossdiffusion term of the flux at the control-volume face. A simple two-dimensional isotropic diffusion equation for which an analytical solution is available is chosen for benchmarking purposes.