A direct O(Nlog2N) finite difference method for fractional diffusion equations

被引：301

作者：

Wang, Hong ^{[1
,2
]}

Wang, Kaixin ^{[1
]}

Sircar, Treena ^{[2
]}

机构：

[1] Shandong Univ, Sch Math, Jinan 250100, Shandong, Peoples R China

[2] Univ S Carolina, Dept Math, Columbia, SC 29208 USA

来源：

JOURNAL OF COMPUTATIONAL PHYSICS | 2010年 / 229卷 / 21期

基金：

美国国家科学基金会;

关键词：

Anomalous diffusion; Circulant and Toeplitz matrices; Fast finite difference methods; Fast Fourier transform; Fractional diffusion equations; NUMERICAL APPROXIMATION; SPACE; TRANSPORT; STABILITY;

D O I：

10.1016/j.jcp.2010.07.011

中图分类号：

TP39 [计算机的应用];

学科分类号：

081203 ; 0835 ;

摘要：

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second-order diffusion equations. Because of the non-local property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O(N-2) and computational cost of O(N-3) where N is the number of grid points. In this paper we develop a fast finite difference method for fractional diffusion equations, which only requires storage of O(N) and computational cost of O( Nlog(2)N) while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method. (C) 2010 Elsevier Inc. All rights reserved.

引用

页码：8095 / 8104

页数：10

共 39 条

[11] Variational formulation for the stationary fractional advection dispersion equation
Ervin, VJ
Roop, JP
[J]. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, 2006, 22 (03) : 558 - 576
[12] Toeplitz and Circulant Matrices: A Review
Gray, Robert M.
[J]. FOUNDATIONS AND TRENDS IN COMMUNICATIONS AND INFORMATION THEORY, 2006, 2 (03): : 155 - 239
[13] HILFER R, 2000, APPLICATIONS FRACTIO
[14] Fractal stream chemistry and its implications for contaminant transport in catchments
Kirchner, JW
Feng, XH
Neal, C
[J]. NATURE, 2000, 403 (6769) : 524 - 527
[15] The accuracy and stability of an implicit solution method for the fractional diffusion equation
Langlands, TAM
Henry, BI
[J]. JOURNAL OF COMPUTATIONAL PHYSICS, 2005, 205 (02) : 719 - 736
[16] Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation
Li, Xianjuan
Xu, Chuanju
[J]. COMMUNICATIONS IN COMPUTATIONAL PHYSICS, 2010, 8 (05) : 1016 - 1051
[17] Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation
Lin, R.
Liu, F.
Anh, V.
Turner, I.
[J]. APPLIED MATHEMATICS AND COMPUTATION, 2009, 212 (02) : 435 - 445
[18] Finite difference/spectral approximations for the time-fractional diffusion equation
Lin, Yumin
Xu, Chuanju
[J]. JOURNAL OF COMPUTATIONAL PHYSICS, 2007, 225 (02) : 1533 - 1552
[19] Numerical solution of the space fractional Fokker-Planck equation
Liu, F
Anh, V
Turner, I
[J]. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2004, 166 (01) : 209 - 219
[20] Magin R.L., 2006, Fractional calculus in bioengineering

← 1 2 3 4 →