Crank-Nicolson Alternative Direction Implicit Method for Space-Fractional Diffusion Equations with Nonseparable Coefficients

In this paper, we study the Crank--Nicolson alternative direction implicit (ADI) method for two-dimensional Riesz space-fractional diffusion equations with nonseparable coefficients. Existing ADI methods are only shown to be unconditional stable when coefficients are some special separable functions. The main contribution of this paper is to show under mild assumptions the unconditional stability of the proposed Crank--Nicolson ADI method in discrete norm and the consistency of cross perturbation terms arising from the Crank--Nicolson ADI method. Also, we demonstrate that several consistent spatial discretization schemes satisfy the required assumptions. Numerical results are presented to examine the accuracy and the efficiency of the proposed ADI methods.