A High-Order Accuracy Method for Solving the Fractional Diffusion Equations

A High-Order Accuracy Method for Solving the Fractional Diffusion Equations

Year:    2020

Author:    Maohua Ran, Chengjian Zhang

Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 239–253

Abstract

In this paper, an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods. In order to efficiently solve the generated linear large-scale system, the generalized minimal residual (GMRES) algorithm is 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 higher order accuracy both in space and in time than the existing methods. When the used boundary value method is $A_{k1,k2}$-stable, it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1. It implies that the iterative solution is convergent rapidly. Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1805-m2017-0081

Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 239–253

Published online:    2020-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    15

Keywords:    Boundary value method Circulant preconditioner High accuracy Generalized Dirichlet type boundary condition.

Author Details

Maohua Ran

Chengjian Zhang