@Article{AAMM-11-4,
author = {Zhiyong, Xing and Wen, Liping},
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 = {<p style="text-align: justify;">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 $h_{x},$ $h_{y}$&nbsp; 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.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.OA-2018-0162},
url = {https://global-sci.com/article/73138/the-fast-implementation-of-the-adi-cn-method-for-a-class-of-two-dimensional-riesz-space-fractional-diffusion-equations}
}