Volume 11, Issue 4
The Fast Implementation of the ADI-CN Method for a Class of Two-Dimensional Riesz Space-Fractional Diffusion Equations

Adv. Appl. Math. Mech., 11 (2019), pp. 942-956.

Published online: 2019-06

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• 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.

• Keywords

Space-fractional diffusion equation, Riesz fractional derivative, alternating direction method, convergence and stability, $L_{\infty}$-norm.

35R11, 65M06, 65M12

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@Article{AAMM-11-942, author = {}, title = {The Fast Implementation of the ADI-CN Method for a Class of Two-Dimensional Riesz Space-Fractional Diffusion Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {4}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0162}, url = {http://global-sci.org/intro/article_detail/aamm/13195.html} }
TY - JOUR T1 - The Fast Implementation of the ADI-CN Method for a Class of Two-Dimensional Riesz Space-Fractional Diffusion Equations JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 942 EP - 956 PY - 2019 DA - 2019/06 SN - 11 DO - http://dor.org/10.4208/aamm.OA-2018-0162 UR - https://global-sci.org/intro/article_detail/aamm/13195.html KW - Space-fractional diffusion equation, Riesz fractional derivative, alternating direction method, convergence and stability, $L_{\infty}$-norm. AB -

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.

Zhiyong Xing & Liping Wen. (2019). The Fast Implementation of the ADI-CN Method for a Class of Two-Dimensional Riesz Space-Fractional Diffusion Equations. Advances in Applied Mathematics and Mechanics. 11 (4). 942-956. doi:10.4208/aamm.OA-2018-0162
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