An ADI Iteration Method for Solving Discretized Two-Dimensional Space-Fractional Diffusion Equations

The two-dimensional (2D) space-fractional diffusion equations can be effectively discretized by an implicit finite difference scheme with the shifted Gr & uuml;nwald formula. The coefficient matrices of the discretized linear systems are equal to the sum of the identity matrix and a block-Toeplitz with a Toeplitz-block matrix. In this paper, one variant of the alternating direction implicit (ADI) iteration method is proposed to solve the discretized linear systems. By making use of suitable permutations, each iteration of the ADI iteration method requires the solutions of two linear subsystems whose coefficient matrices are block diagonal matrices with diagonal blocks being Toeplitz matrices. These two linear subsystems can be solved block by block by fast or superfast direct methods. Theoretical analyses show that the ADI iteration method is convergent. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduce the optimal value of its iteration parameter. Numerical results exhibit that the corresponding ADI preconditioner can improve the computational efficiency of the Krylov subspace iteration methods.