TY - JOUR
T1 - Using RBF-Generated Quadrature Rules to Solve Nonlocal Anomalous Diffusion
AU - Lyngaas , Isaac
AU - Peterson , Janet
JO - International Journal of Numerical Analysis and Modeling
VL - 4-5
SP - 628
EP - 648
PY - 2018
DA - 2018/04
SN - 15
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/12535.html
KW - Nonlocal, anomalous diffusion, radial basis functions, RBF, quadrature.
AB - <p style="text-align: justify;">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.</p>