Adv. Appl. Math. Mech., 11 (2019), pp. 942-956.
Published online: 2019-06
Cited by
- BibTex
- RIS
- TXT
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} }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.