A High-Order Accuracy Method for Solving the Fractional Diffusion Equations (Journal of Computational Mathematics, 2020)
In this paper, an efficient numerical method for solving the general fractional diffusionequations with Riesz fractional derivative is proposed by combining the fractional compactdifference operator and the boundary value methods. In order to efficiently solve thegenerated linear large-scale system, the generalized minimal residual (GMRES) algorithmis applied. For accelerating the convergence rate of the iterative, the Strang-type, Chan-type and P-type preconditioners are introduced. The suggested method can reach higherorder accuracy both in space and in time than the existing methods. When the usedboundary value method is $A_{k1,k2}$-stable, it is proven that Strang-type preconditioner isinvertible and the spectra of preconditioned matrix is clustered around 1. It implies thatthe iterative solution is convergent rapidly. Numerical experiments with the absorbingboundary condition and the generalized Dirichlet type further verify the efficiency.