Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations (Communications in Computational Physics, 2018)
In this paper, preconditioned iterative methods for solving two-dimensionalspace-fractional diffusion equations are considered. The fractional diffusion equationis discretized by a second-order finite difference scheme, namely, the Crank-Nicolsonweighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian,H. Zhou and W. Deng, A class of second order difference approximation for solving spacefractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretizedlinear systems, we first propose preconditioned iterative methods to solve them. Thenwe apply the D'Yakonov ADI scheme to split the linear systems and solve the obtainedsplitting systems by iterative methods. Two preconditioned iterative methods, the preconditionedgeneralized minimal residual (preconditioned GMRES) method and thepreconditioned conjugate gradient normal residual (preconditioned CGNR) method,are proposed to solve relevant linear systems. By fully exploiting the structure of thecoefficient matrix, we design two special kinds of preconditioners, which are easilyconstructed and are able to accelerate convergence of iterative solvers. Numericalresults show the efficiency of our preconditioners.