On the use of surface interpolation techniques in generalised finite volume strategies for simulating transport in highly anisotropic porous media

A control volume technique for solving a representative diffusion equation in an orthotropic medium is considered. The approximation of the cross-diffusion flux term is of utmost important for the accuracy of the solution. A preliminary investigation that used exact function values from an available analytical solution to approximate this term during the numerical simulation provided excellent agreement with the exact solution. This finding motivated the need for accurate surface interpolation techniques for estimating the cross-diffusion term. The use of radial basis functions is a well-known interpolation technique for fitting scattered data, which can be considered as a global interpolating method because function values in the whole solution domain contribute towards the interpolation. A number of radial basis functions (RBF) was used to approximate the gradients in the cross-diffusion flux term and it was found that the accuracy of the finite volume solution was generally poor. It was concluded that the RBF estimated function does not reflect local variation of the solution, particularly for the gradients. Another strategy for local function estimation concerns the weighted least-squares method. Different variants of this method were analysed here for approximating the cross-diffusion term and it was found that the numerical results well matched the exact solution. The results highlight that the development of an accurate, generalised finite volume strategy requires a highly accurate flux approximation to enable second-order spatial accuracy to be achieved.