Year: 2022
Author: Li-Lian Wang, Huiyuan Li, Ruiqing Liu, Li-Lian Wang
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 1009–1040
Abstract
In this paper, we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains. We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian. As a result, the “stiffness” matrix can be fast computed and assembled via the four-point stable recursive algorithm with $\mathcal{O}(N^2)$ arithmetic operations. Moreover, the singular factor in a typical kernel function can be fully absorbed by the basis. With the aid of Fourier analysis, we can prove the convergence of the scheme. We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions. We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2022-0007s
Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 4 : pp. 1009–1040
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Nonlocal diffusion equation spectral-Galerkin Hermite functions correlation/convolution recurrence algorithm.
Author Details
-
A Grid-Overlay Finite Difference Method for the Fractional Laplacian on Arbitrary Bounded Domains
Huang, Weizhang | Shen, JinyeSIAM Journal on Scientific Computing, Vol. 46 (2024), Iss. 2 P.A744
https://doi.org/10.1137/23M1558562 [Citations: 1] -
Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian
Shen, Jinye | Shi, Bowen | Huang, WeizhangCommunications on Applied Mathematics and Computation, Vol. (2024), Iss.
https://doi.org/10.1007/s42967-024-00368-z [Citations: 0] -
A sparse spectral method for fractional differential equations in one-spatial dimension
Papadopoulos, Ioannis P. A. | Olver, SheehanAdvances in Computational Mathematics, Vol. 50 (2024), Iss. 4
https://doi.org/10.1007/s10444-024-10164-1 [Citations: 0]