A Second Order Finite Volume Technique for Simulating Transport in Anisotropic Media

An existing two-dimensional finite volume technique is modified by introducing a correction term to increase the accuracy of the method to second order. It is well known that the accuracy of the finite volume method strongly depends on the order of the approximation of the flux term at the control volume (CV) faces. For highly orthotropic and anisotropic media, first order approximations produce inaccurate simulation results, which motivates the need for better estimates of the flux expression. In this article, a new approach to approximate the flux term at the CV face is presented. The discretisation involves a decomposition of the flux and an improved least squares approximation technique to calculate the derivatives of the dependent function on the CV faces for estimating both the cross diffusion term and a correction for the primary flux term. The advantage of this method is that any arbitrary unstructured mesh can be used to implement the technique without considering the shapes of the mesh elements. It was found that the numerical results well matched the available exact solution for a representative transport equation in highly orthotropic media and the benchmark solutions obtained on a fine mesh for anisotropic media. Previously proposed CV techniques are compared with the new method to highlight its accuracy for different unstructured meshes

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 di3usion equation in an orthotropic medium is considered.The approximation of the cross-di3usion 6ux 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 &nding motivated the need for accurate surface interpolation techniques for estimating the cross-di3usion term.The use of radial basis functions is a well-known interpolation technique for &tting 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-di3usion 6ux term and it was found that the accuracy of the &nite volume solution was generally poor.It was concluded that the RBF estimated function does not re6ect local variation of the solution, particularly for the gradients.Another strategy for local function estimation concerns the weighted least-squares method.Di3erent variants of this method were analysed here for approximating the cross-di3usion term and it was found that the numerical results well matched the exact solution.The results highlight that the development of an accurate, generalised &nite volume strategy requires a highly accurate 6ux approximation to enable second-order spatial accuracy to be achieved

A Second Order Control-Volume Finite-Element Least-Squares Strategy for Simulating Diffusion in Strongly Anisotropic Media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality

Generalised Finite Volume Strategies for Simulating Transport in Strongly Orthotropic Porous Media

In this work two different finite volume computational strategies for solving a representative two-dimensional diffusion equation in an orthotropic medium are considered. When the diffusivity tensor is treated as linear, this problem admits an analytic solution used for analysing the accuracy of the proposed numerical methods. In the first method, the gradient approximation techniques discussed by Jayantha and Turner [Numerical Heat Transfer, Part B: Fundamentals, 40, pp.367--390, 2001] are applied directly to the diffusion equation. In the second method, the diffusion equation is transformed via scaling parameters to an isotropic model and then the control volume techniques discussed by Jayantha and Turner are used to obtain the numerical results on the transformed domain. Both methods are shown to produce reasonable results in comparison with the exact solution for a range of anisotropy ratios typical of wood. However, only the first method is appropriate for use in non-linear coupled transport systems. This work highlights the necessity of determining a higher order gradient approximation to improve the numerical results on the untransformed domain

A Comparison of Gradient Approximations for Use in Finite-Volume Computational Models for Two-Dimensional Diffusion Equations

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