Using RBF-Generated Quadrature Rules to Solve Nonlocal Anomalous Diffusion

Abstract

The goal of this work is to solve nonlocal diffusion and anomalous diffusion problems by approximating the nonlocal integral appearing in the integro-differential equation by novel quadrature rules. These quadrature rules are derived so that they are exact for a nonlocal integral evaluated at translations of a given radial basis function (RBF). We first illustrate how to derive RBF-generated quadrature rules in one dimension and demonstrate their accuracy for approximating a nonlocal integral. Once the quadrature rules are derived as a preprocessing step, we apply them to approximate the nonlocal integral in a nonlocal diffusion problem and when the temporal derivative is approximated by a standard difference approximation a system of difference equations are obtained. This approach is extended to two dimensions where both a circular and rectangular nonlocal neighborhood are considered. Numerical results are provided and we compare our results to published results solving nonlocal problems using standard finite element methods.