A fast preconditioning iterative method for solving the discretized second-order space-fractional advection–diffusion equations

Abstract

Space-fractional advection–diffusion equations (SFADEs) arise in many application areas. In this paper, we propose fast second-order numerical methods for solving the one- and two-dimensional SFADEs defined on a finite domain. The Crank–Nicolson difference scheme is utilized to discretize the temporal derivatives, the weighted and shifted Grünwald difference operators are employed to discretize the spatial fractional derivatives in SFADEs. We analyze the stability and convergence of the difference schemes by using the matrix analysis method. The coefficient matrix of the discretized system of linear equations has the structure of the sum of an identity matrix and non-symmetric Toeplitz matrices. New τ-matrix approximate preconditioners are proposed for the discretized system of linear equations for both one- and two-dimensional SFADEs, respectively. The generalized minimal residual (GMRES) methods combined with the proposed preconditioners are applied to solve the linear systems. We analyze the convergence rate of the GMRES method for solving the preconditioned linear systems. Numerical results demonstrate the effectiveness of the proposed methods.