The Fast Implementation of the ADI-CN Method for a Class of Two-Dimensional Riesz Space-Fractional Diffusion Equations
Year: 2019
Author: Zhiyong Xing, Liping Wen
Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 4 : pp. 942–956
Abstract
In this paper, a class of two-dimensional Riesz space-fractional diffusion equations (2D-RSFDE) with homogeneous Dirichlet boundary conditions is considered. In order to reduce the computational complexity, the alternating direction implicit Crank-Nicholson (ADI-CN) method is applied to reduce the two-dimensional problem into a series of independent one-dimensional problems. Based on the fact that the coefficient matrices of these one-dimensional problems are all real symmetric positive definite Toeplitz matrices, a fast method is developed for the implementation of the ADI-CN method. It is proved that the ADI-CN method is uniquely solvable, unconditionally stable and convergent with order $\mathcal{O}(\tau^{2}+h_{x}^{2}+h_{y}^{2})$ in the discrete $L_{\infty}$-norm with time step $\tau$ and mesh size $h_{x},$ $h_{y}$ in the $x$ direction and the $y$ direction, respectively. Finally, several numerical results are provided to verify the theoretical results and the efficiency of the fast method.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2018-0162
Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 4 : pp. 942–956
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Space-fractional diffusion equation Riesz fractional derivative alternating direction method convergence and stability $L_{\infty}$-norm.
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