Using RBF-Generated Quadrature Rules to Solve Nonlocal Continuum Models

Recently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating time-dependent nonlocal integro-differential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due to the approximation of the nonlocal integral. In this work, we propose a novel method based on using radial basis functions to generate accurate quadrature rules for the nonlocal integral appearing in the model and then coupling this with a finite difference approximation to the time-dependent terms. The viability of our method is demonstrated through various numerical tests on time dependent nonlocal diffusion, nonlocal anomalous diffusion, and nonlocal advection problems in one and two dimensions. In addition to nonlocal problems with continuous solutions, we modify our approach to handle problems with discontinuous solutions. We compare some numerical results with analogous finite element results and demonstrate that for an equivalent amount of computational work we obtain much higher rates of convergence.