An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients

doi:10.1007/s42967-019-00050-9

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (2): 215-239.doi: 10.1007/s42967-019-00050-9

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An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients

Xue-lei Lin¹, Pin Lyu², Michael K. Ng³, Hai-Wei Sun⁴, Seakweng Vong⁴

1 Department of Mathematics, Hong Kong Baptist University, Hong Kong, China;
2 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China;
3 Department of Mathematics, The University of Hong Kong, Hong Kong, China;
4 Department of Mathematics, University of Macau, Macao, China

收稿日期:2019-06-02 修回日期:2019-10-11 出版日期:2020-06-20 发布日期:2020-02-19
通讯作者: Pin Lyu, Xue-lei Lin, Michael K. Ng, Hai-Wei Sun, Seakweng Vong E-mail:plyu@swufe.edu.cn;hxuellin@gmail.com;mng@maths.hku.hk;hsun@um.edu.mo;swvong@um.edu.mo
基金资助:
This research was supported by research Grants, 12306616, 12200317, 12300519, 12300218 from HKRGC GRF, 11801479 from NSFC, MYRG2018-00015-FST from University of Macau, and 0118/2018/A3 from FDCT of Macao, Macao Science and Technology Development Fund 0005/2019/A, 050/2017/A, and the Grant MYRG2017-00098-FST and MYRG2018-00047-FST from University of Macau.

An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients

Xue-lei Lin¹, Pin Lyu², Michael K. Ng³, Hai-Wei Sun⁴, Seakweng Vong⁴

1 Department of Mathematics, Hong Kong Baptist University, Hong Kong, China;
2 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China;
3 Department of Mathematics, The University of Hong Kong, Hong Kong, China;
4 Department of Mathematics, University of Macau, Macao, China

Received:2019-06-02 Revised:2019-10-11 Online:2020-06-20 Published:2020-02-19
Contact: Pin Lyu, Xue-lei Lin, Michael K. Ng, Hai-Wei Sun, Seakweng Vong E-mail:plyu@swufe.edu.cn;hxuellin@gmail.com;mng@maths.hku.hk;hsun@um.edu.mo;swvong@um.edu.mo
Supported by:
This research was supported by research Grants, 12306616, 12200317, 12300519, 12300218 from HKRGC GRF, 11801479 from NSFC, MYRG2018-00015-FST from University of Macau, and 0118/2018/A3 from FDCT of Macao, Macao Science and Technology Development Fund 0005/2019/A, 050/2017/A, and the Grant MYRG2017-00098-FST and MYRG2018-00047-FST from University of Macau.

摘要/Abstract

摘要： In this paper, a second-order fnite-diference scheme is investigated for time-dependent space fractional difusion equations with variable coefcients. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefcients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efciency of the proposed scheme.

关键词: One-side space fractional difusion equation, Variable difusion coefcients, Stability and convergence, High-order fnite-diference scheme, Preconditioner

Abstract: In this paper, a second-order fnite-diference scheme is investigated for time-dependent space fractional difusion equations with variable coefcients. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefcients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efciency of the proposed scheme.

Key words: One-side space fractional difusion equation, Variable difusion coefcients, Stability and convergence, High-order fnite-diference scheme, Preconditioner

中图分类号:

Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong. An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 215-239.

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An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients

An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients

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