A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

doi:10.1007/s42967-020-00069-3

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (1): 157-176.doi: 10.1007/s42967-020-00069-3

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A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

Jun-Feng Yin, Yi-Shu Du

School of Mathematical Sciences, Tongji University, Shanghai 200092, China

Received:2019-09-30 Revised:2020-03-18 Online:2021-03-20 Published:2021-03-15
Contact: Jun-Feng Yin, yinjf@tongji.edu.cn;Yi-Shu Du, duyishu@tongji.edu.cn E-mail:yinjf@tongji.edu.cn;duyishu@tongji.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (No. 11971354). The author Yi-Shu Du acknowledges the financial support from the China Scholarship Council (File No. 201906260146).

Abstract

Abstract: After discretization by the finite volume method, the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure. The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix. Moreover, we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations, which is unconditionally convergent for any positive constant. Meanwhile, the iteration methods introduce a new preconditioner for Krylov subspace methods. Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner, compared with the existing approaches.

Key words: Fractional diffusion equations, Finite volume method, Operator splitting, Positive-definite

CLC Number:

Jun-Feng Yin, Yi-Shu Du. A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 157-176.

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A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

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