The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation

CHEN Jian; ZENG Taishan

doi:10.6054/j.jscnun.2020051

Journal of South China Normal University (Natural Science Edition) > 2020 > 52(3): 106-110. > DOI: 10.6054/j.jscnun.2020051

CHEN Jian, ZENG Taishan. The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation[J]. Journal of South China Normal University (Natural Science Edition), 2020, 52(3): 106-110. DOI: 10.6054/j.jscnun.2020051

Citation:

PDF (1039 KB)

The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation

CHEN Jian¹,
ZENG Taishan^2, ,

1.
School of Mathematics and Big Data, Foshan University, Foshan 528000, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

More Information

Received Date: January 13, 2020
Available Online: March 21, 2021

Graphical Abstract

Abstract

Abstract

Based on L1 formula and the multiscale Galerkin method, a fully-discrete scheme is proposed for solving time fractional subdiffusion equations with α order Caputo fractional derivative. The existence and uniqueness of the solution of the fully-discrete scheme are proved, and the optimal convergence order O(h^r+ τ ^2-α) is also deduced, where r is the order of piecewise polynomials. A multilevel augmentation method (MAM) is developed to solve the linear systems resulting from the fully-discrete scheme at each time step, and MAM preserves the optimal convergence order. A numerical experiment is presented at last to show the validity of the theoretical analysis.
- L1 approximation,
- multiscale orthogonal base,
- multilevel augmentation method,
- fractional subdiffusion equation,
- convergence order

FullText(HTML)

References (15)

References

[1]	KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M].Amsterdam:Elsevier, 2006.
[2]	刘发旺, 庄平辉, 刘青霞.分数阶偏微分方程数值方法及其应用[M].北京:科学出版社, 2015.
[3]	ZHUANG P H, LIU F W. Implicit difference approximation for the time fractional diffusion equation[J]. Journal of Applied Mathematics Computing, 2006, 22(3):87-99. doi: 10.1007-BF02832039/
[4]	JI C C, SUN Z Z. The high-order compact numerical algorithms for the two-dimensional fractional subdiffusion equation[J]. Applied Mathematics and Computation, 2015, 269:775-791. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=6d7eedc38b49765c63fe86f73fe4c7bd
[5]	REN J C, MAO S P, ZHANG J W. Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation[J]. Numerical Methods for Partial Differential Equations, 2018, 34(2):705-730. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=10.1002/num.22226
[6]	ZENG F H, LI C P, LIU F W, et al. Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy[J]. SIAM Journal on Scientific Computing, 2015, 37(1):A55-A78. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=b0ca5d268d6264c9f060cf12e172e31b
[7]	ROOP J P. Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in $\mathbb{R}$ ²[J]. Journal of Computational and Applied Mathematics, 2006, 193:243-268.
[8]	ZHAO Z G, LI C P. Fractional difference/finite element approximations for the time space fractional telegraph equation[J]. Applied Mathematics and Computation, 2012, 219:2975-2988. http://cn.bing.com/academic/profile?id=1d9585f5b67028738e127b90e64c1df5&encoded=0&v=paper_preview&mkt=zh-cn
[9]	CARELLA A R, DORAO C A. Least-squares spectral method for the solution of a fractional advection-dispersion equation[J]. Journal of Computational Physics, 2013, 232:33-45. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=34d74032c04e005befd4b78e31549a89
[10]	LI X J, XU C J. A space-time spectral method for the time fractional diffusion equation[J]. SIAM Journal on Numerical Analysis, 2009, 47:2108-2131. http://cn.bing.com/academic/profile?id=45c4c1b0fd2ada0312bc53ac5d01f303&encoded=0&v=paper_preview&mkt=zh-cn
[11]	PEDAS A, TAMME E. On the convergence of spline co-llocation methods for solving fractional differential equations[J]. Journal of Computational and Applied Mathematics, 2011, 235(12):3502-3514. https://www.sciencedirect.com/science/article/pii/S0377042711000823
[12]	PODLUBNY I, CHECHKIN A, SKOVRANEK T, et al. Matrix approach to discrete fractional calculus Ⅱ:partial fractional differential equations[J]. Journal of Computational Physics, 2009, 228:3137-3153. http://d.old.wanfangdata.com.cn/OAPaper/oai_arXiv.org_0811.1355
[13]	CHEN Z Y, WU B, XU Y S. Multilevel augmentation me-thods for operator equations[J]. Numerical Mathematics:A Journal of Chinese Universities, 2005, 14:31-55.
[14]	CHEN Z Y, WU B, XU Y S. Multilevel augmentation me-thods for differential equations[J]. Advances in Computational Mathematics, 2006, 24:213-238. https://www.researchgate.net/publication/220390842_Multilevel_augmentation_methods_for_differential_equations
[15]	CHEN Z Y, MICCHELLI C A, XU Y S. Multiscale me-thods for Fredholm integral equations[M]. Cambridge:Cambridge University Press, 2015.

Cited By

Cited by

Periodical cited type(2)

1.	李宪，达举霞，章欢. 四阶两点边值问题n个对称正解的存在性. 华南师范大学学报(自然科学版). 2024(01): 123-127 .
2.	达举霞. 四阶两点边值问题3个对称正解的存在性. 华南师范大学学报(自然科学版). 2021(01): 90-93 .

Other cited types(0)

Get Citation

PDF

XML

Article views (1459) PDF downloads (91) Cited by(2)

Turn off MathJax

Article Contents

Abstract

References

The Multilevel Augmentation Method for Solving Time Fractional Subdiffusion Equation

Abstract

References

Cited by

Periodical cited type(2)

Other cited types(0)

Catalog

Export File

Citation

Format

Content