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Approximating Nonlocal Diffusion Problems Using Quadrature Rules Generated by Radial Basis Functions

Title: Approximating Nonlocal Diffusion Problems Using Quadrature Rules Generated by Radial Basis Functions.
Name(s): Lyngaas, Isaac Ron, author
Peterson, Janet S., professor directing thesis
Gunzburger, Max D., committee member
Burkardt, John V., committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Mathematics, degree granting department
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2016
Publisher: Florida State University
Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (80 pages)
Language(s): English
Abstract/Description: Nonlocal models differ from traditional partial differential equation (PDE) models because they contain no spatial derivatives; instead an appropriate integral is used. Nonlocal models are especially useful in the case where there are issues calculating the spatial derivatives of a PDE model. In many applications (e.g., biological systems, flow through porous media) the observed rate of diffusion is not accurately modeled by the standard diffusion differential operator but rather exhibits so-called anomalous diffusion. Anomalous diffusion can be represented in a PDE model by using a fractional Laplacian operator in space whereas the nonlocal approach only needs to slightly modify its integral formulation to model anomalous diffusion. Anomalous diffusion is one such case where approximating the spatial derivative operator is a difficult problem. In this work, an approach for approximating standard and anomalous nonlocal diffusion problems using a new technique that utilizes radial basis functions (RBFs) is introduced and numerically tested. The typical approach for approximating nonlocal diffusion problems is to use a Galerkin formulation. However, the Galerkin formulation for nonlocal diffusion problems can often be difficult to compute efficiently and accurately especially for problems in multiple dimensions. Thus, we investigate the alternate approach of using quadrature rules generated by RBFs to approximate the nonlocal diffusion problem. This work will be split into three major parts. The first will introduce RBFs and give some examples of how they are used. This part will motivate our approach for using RBFs on the nonlocal diffusion problem. In the second part, we will derive RBF-generated quadrature rules in one dimension and show they can be used to approximate nonlocal diffusion problems. The final part will address how the RBF quadrature approach can be extended to higher dimensional problems. Numerical test cases are shown for both the standard and anomalous nonlocal diffusion problems and compared with standard finite element approximations. Preliminary results show that the method introduced is viable for approximating nonlocal diffusion problems and that highly accurate approximations are possible using this approach.
Identifier: FSU_FA2016_Lyngaas_fsu_0071N_13512 (IID)
Submitted Note: A Thesis submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Master of Science.
Degree Awarded: Fall Semester 2016.
Date of Defense: September 9, 2016.
Keywords: Anomalous Diffusion, Nonlocal Model, Quadrature Rule, Radial Basis Function
Bibliography Note: Includes bibliographical references.
Advisory Committee: Janet Peterson, Professor Directing Thesis; Max Gunzburger, Committee Member; John Burkardt, Committee Member.
Subject(s): Applied mathematics
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Host Institution: FSU

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Lyngaas, I. R. (2016). Approximating Nonlocal Diffusion Problems Using Quadrature Rules Generated by Radial Basis Functions. Retrieved from