A compact finite difference scheme for the fractional sub-diffusion equations

被引：353

作者：

Gao, Guang-hua ^{[1
]}

Sun, Zhi-zhong ^{[1
]}

机构：

[1] Southeast Univ, Dept Math, Nanjing 211189, Peoples R China

来源：

JOURNAL OF COMPUTATIONAL PHYSICS | 2011年 / 230卷 / 03期

基金：

中国国家自然科学基金;

关键词：

Fractional sub-diffusion equation; L1; discretization; Compact scheme; Stability; Convergence; Energy method; ANOMALOUS SUBDIFFUSION EQUATION; IMPLICIT NUMERICAL-METHOD; STABILITY; APPROXIMATIONS; DISPERSION; ACCURACY; ORDER;

D O I：

10.1016/j.jcp.2010.10.007

中图分类号：

TP39 [计算机的应用];

学科分类号：

081203 ; 0835 ;

摘要：

In this paper, a compact finite difference scheme for the fractional sub-diffusion equations is derived. After a transformation of the original problem, the L1 discretization is applied for the time-fractional part and fourth-order accuracy compact approximation for the second-order space derivative. The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme in maximum norm are proved using the energy method, where a new inner product is introduced for the theoretical analysis. The technique is quite novel and different from previous analytical methods. Finally, a numerical example is provided to show the effectiveness and accuracy of the method. (C) 2010 Elsevier Inc. All rights reserved.

引用

页码：586 / 595

页数：10

共 26 条

[1] [Anonymous], 2006, THEORY APPL FRACTION
[2] [Anonymous], ANOMALOUS TRANSPORT
[3] [Anonymous], 1999, FRACTIONAL DIFFERENT
[4] Fractional dispersion, Levy motion, and the MADE tracer tests
Benson, DA
Schumer, R
Meerschaert, MM
Wheatcraft, SW
[J]. TRANSPORT IN POROUS MEDIA, 2001, 42 (1-2) : 211 - 240
[5] Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation
Chen, Chang-ming
Liu, F.
Burrage, K.
[J]. APPLIED MATHEMATICS AND COMPUTATION, 2008, 198 (02) : 754 - 769
[6] A Fourier method for the fractional diffusion equation describing sub-diffusion
Chen, Chang-Ming
Liu, F.
Turner, I.
Anh, V.
[J]. JOURNAL OF COMPUTATIONAL PHYSICS, 2007, 227 (02) : 886 - 897
[7] NUMERICAL SCHEMES WITH HIGH SPATIAL ACCURACY FOR A VARIABLE-ORDER ANOMALOUS SUBDIFFUSION EQUATION
Chen, Chang-Ming
Liu, F.
Anh, V.
Turner, I.
[J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2010, 32 (04) : 1740 - 1760
[8] Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation
Chen, Chang-Ming
Liu, Fawang
Turner, Ian
Anh, Vo
[J]. NUMERICAL ALGORITHMS, 2010, 54 (01) : 1 - 21
[9] Finite difference approximations for the fractional Fokker-Planck equation
Chen, S.
Liu, F.
Zhuang, P.
Anh, V.
[J]. APPLIED MATHEMATICAL MODELLING, 2009, 33 (01) : 256 - 273
[10] Compact finite difference method for the fractional diffusion equation
Cui, Mingrong
[J]. JOURNAL OF COMPUTATIONAL PHYSICS, 2009, 228 (20) : 7792 - 7804

← 1 2 3 →