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, 2019)
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 $\mathcal{O}(τ^{2}+h_{x}^{2}+h_{y}^{2})$ in the discrete $L_{∞}$-norm with time step $τ$ and mesh size $h_{x},$ $h_{y}$\u00a0 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.