A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

doi:10.4208/jcm.1805-m2017-0081

Journal of Computational Mathematics ›› 2020, Vol. 38 ›› Issue (2): 239-253.DOI: 10.4208/jcm.1805-m2017-0081

A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

Maohua Ran^1,2, Chengjian Zhang³

1. School of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China;
2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2017-03-27 Revised:2017-09-19 Online:2020-03-15 Published:2020-03-15
Supported by:
The authors would like to thank the referee and the editors for their valuable comments and suggestions, which have let to great improvements over the original manuscript. This work was supported in part by National Natural Science Foundation of China under grants 11801389 and 11571128.

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Maohua Ran, Chengjian Zhang. A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS[J]. Journal of Computational Mathematics, 2020, 38(2): 239-253.

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, Chantype 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.

Key words: Boundary value method, Circulant preconditioner, High accuracy, Generalized Dirichlet type boundary condition

CLC Number:

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A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

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