Volume 38, Issue 2
A High-Order Accuracy Method for Solving the Fractional Diffusion Equations

J. Comp. Math., 38 (2020), pp. 239-253.

Published online: 2020-02

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• 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.

65F08, 65M06, 65M12

maohuaran@163.com (Maohua Ran)

cjzhang@hust.edu.cn (Chengjian Zhang)

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@Article{JCM-38-239, author = {Ran , Maohua and Zhang , Chengjian}, title = {A High-Order Accuracy Method for Solving the Fractional Diffusion Equations}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1805-m2017-0081}, url = {http://global-sci.org/intro/article_detail/jcm/14516.html} }
TY - JOUR T1 - A High-Order Accuracy Method for Solving the Fractional Diffusion Equations AU - Ran , Maohua AU - Zhang , Chengjian JO - Journal of Computational Mathematics VL - 2 SP - 239 EP - 253 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1805-m2017-0081 UR - https://global-sci.org/intro/article_detail/jcm/14516.html KW - Boundary value method, Circulant preconditioner, High accuracy, Generalized Dirichlet type boundary condition. AB -

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.

Maohua Ran & Chengjian Zhang. (2020). A High-Order Accuracy Method for Solving the Fractional Diffusion Equations. Journal of Computational Mathematics. 38 (2). 239-253. doi:10.4208/jcm.1805-m2017-0081
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