Volume 11, Issue 4
The Fast Implementation of the ADI-CN Method for a Class of Two-Dimensional Riesz Space-Fractional Diffusion Equations
10.4208/aamm.OA-2018-0162

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

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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.

• History

Published online: 2019-06