(untitled) (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 O(τ^2+h_x^2+h_y^2) in the discrete L_∞-norm with time step τ and mesh size h_x, h_y00a0 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.