On ADI-like iteration method for fractional diffusion equations

The fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of the identity matrix with two diagonal-times-Toeplitz matrices. In this paper, the alternating direction implicit (ADI)-like method based on the classical ADI iteration is proposed to solve the discretized linear system. Theoretical analyses show that the ADI-like iteration method is convergent. Each iteration of this method requires the solutions of two linear systems with Hessenberg coefficient matrices. These two systems can be solved effectively by Hessenberg LU factorization. Numerical examples are presented to illustrate the effectiveness of the ADI-like iteration method.