Year: 2018
Author: Isaac Lyngaas, Janet Peterson
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 4-5 : pp. 628–648
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2018-IJNAM-12535
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 4-5 : pp. 628–648
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Nonlocal anomalous diffusion radial basis functions RBF quadrature.