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Volume 15, Issue 4-5
Using RBF-Generated Quadrature Rules to Solve Nonlocal Anomalous Diffusion

Isaac Lyngaas & Janet Peterson

Int. J. Numer. Anal. Mod., 15 (2018), pp. 628-648.

Published online: 2018-04

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  • 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.

  • Keywords

Nonlocal, anomalous diffusion, radial basis functions, RBF, quadrature.

  • AMS Subject Headings

35R35, 49J40, 60G40

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

irlyngaas@gmail.com (Isaac Lyngaas)

jpeterson@fsu.edu (Janet Peterson)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-15-628, author = {Lyngaas , Isaac and Peterson , Janet}, title = {Using RBF-Generated Quadrature Rules to Solve Nonlocal Anomalous Diffusion}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {4-5}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12535.html} }
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 -

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.

Lyngaas , Isaac and Peterson , Janet. (2018). Using RBF-Generated Quadrature Rules to Solve Nonlocal Anomalous Diffusion. International Journal of Numerical Analysis and Modeling. 15 (4-5). 628-648. doi:
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