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

• ORIGINAL PAPER • 上一篇下一篇

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

收稿日期:2019-09-30 修回日期:2020-03-18 出版日期:2021-03-20 发布日期:2021-03-15
通讯作者: Jun-Feng Yin, yinjf@tongji.edu.cn;Yi-Shu Du, duyishu@tongji.edu.cn E-mail:yinjf@tongji.edu.cn;duyishu@tongji.edu.cn
基金资助:
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).

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).

摘要/Abstract

摘要： 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.

关键词: Fractional diffusion equations, Finite volume method, Operator splitting, Positive-definite

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

中图分类号:

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.

参考文献

1. Bai, Z.-Z., Lu, K.-Y.:On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations. BIT Numer. Math. 59, 1-33 (2018)
2. Bai, Z.-Z., Golub, G.H., Michael, N.K.:Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603-626 (2003)
3. Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.:Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26, 844-863 (2005)
4. Bai, Z.-Z., Yin, J.-F., Su, Y.-F.:A shift-splitting preconditioner for non-Hermitian positive definite matrices. J. Comput. Math. 24, 539-552 (2006)
5. Bai, Z.-Z., Lu, K.-Y., Pan, J.-Y.:Diagonal and Toeplitz splitting iteration methods for diagonal-plusToeplitz linear systems from spatial fractional diffusion equations. Nume. Linear Algebra Appl. 24, 1-15 (2017)
6. Benzi, M., Szyld, D.B.:Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods. Numer. Math. 76, 309-321 (1997)
7. Chan, T.F.:An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Stat. Comput. 9, 766-771 (1988)
8. Donatelli, M., Mazza, M., Serra-Capizzano, S.:Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262-279 (2016)
9. Eisenstat, S.C., Elman, H.C., Schultz, M.H.:Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20, 345-357 (1983)
10. Hestenes, M.R., Stiefel, E.:Methods of conjugate gradients for solving linear. J. Res. Nat. Bur. Stand. 49, 409-436 (1952)
11. Lei, S.-L., Sun, H.-W.:A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715-725 (2013)
12. Pan, J.-Y., Ke, R.-H., Ng, M.K., Sun, H.-W.:Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM J. Sci. Comput. 36, 2698-2719 (2014)
13. Pan, J.-Y., Ng, M.K., Wang, H.:Fast iterative solvers for linear systems arising from time-dependent space-fractional diffusion equations. SIAM J. Sci. Comput. 38, 2806-2826 (2016)
14. Pan, J.-Y., Ng, M., Wang, H.:Fast preconditioned iterative methods for finite volume discretization of steady-state space-fractional diffusion equations. Numer. Algorithms 74, 153-173 (2017)
15. Strang, G.:A proposal for Toeplitz matrix calculations. Stud. Appl. Math. 74, 171-176 (1986)
16. Wang, H., Du, N.:A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations. J. Comput. Phys. 240, 49-57 (2013)
17. Wang, H., Wang, K.-X.:An O(Nlog²N) alternating-direction finite difference method for two-dimensional fractional diffusion equations. J. Comput. Phys. 230, 7830-7839 (2011)
18. Wang, K.-X., Wang, H.:A fast characteristic finite difference method for fractional advection-diffusion equations. Adv. Water Resour. 34, 810-816 (2011)
19. Wang, H., Yang, D.-P.:Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51, 1088-1107 (2013)
20. Wang, H., Wang, K.-X., Sircar, T.:A direct O(Nlog²N) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095-8104 (2010)
21. Yin, J.-F.:A class of preconditioners based on matrix splitting for nonsymmetric linear systems. Appl. Math. Comput. 216, 1694-1706 (2010)
22. Zeleny, M.:Multiple Criteria Decision Making Kyoto 1975, vol. 123. Springer, Berlin (2012)
23. Zheng, N., Hayami, K., Yin, J.-F.:Modulus-type inner outer iteration methods for nonnegative constrained least squares problems. SIAM J. Matrix Anal. Appl. 37, 1250-1278 (2016)

[1]	Qingqing Wu, Zhongshu Wu, Xiaoyan Zeng. A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 509-526.
[2]	Bangti Jin, Zhi Zhou. Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 163-177.
[3]	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.
[4]	Qingqing Wu, Xiaoyan Zeng. Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers' Equations[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 305-318.
[5]	Can Li, Shuming Liu. Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 73-91.
[6]	Yaoqiang Yan, Weihua Deng, Daxin Nie. A Finite-Diference Approximation for the One- and Two-Dimensional Tempered Fractional Laplacian[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 129-145.
[7]	Kamran Kazmi, Abdul Khaliq. A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 525-544.
[8]	Zhiping Mao, Zhen Li, George Em Karniadakis. Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 597-619.
[9]	Yubo Yang, Fanhai Zeng. Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 621-637.

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

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

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 9

编辑推荐

Metrics

本文评价