This paper is concerned with two alternating direction implicit (ADI) finite difference methods for solving a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods is provided in the discrete maximum norm. It is shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. Numerical results are given to confirm the theoretical analysis results. 1. Introduction Fractional differential equations and fractional calculus arise in various application problems in science and engineering [1–16]. Various numerical methods have been developed for the computation of fractional differential equations [17–34]. Fractional subdiffusion equations describe a special type of anomalous diffusion [35], and it is a more difficult task to solve this kind of equation numerically. Numerical works for fractional subdiffusion equations are mostly focused on one-dimensional problems due to the memory effect in fractional derivatives; see, for example, [19, 20, 20–26, 31–33, 36–42]. A two-dimensional anomalous subdiffusion equation was numerically treated in [43, 44], where explicit and implicit finite difference schemes were proposed. Chen et al. [28] extended their work in [43] to a variable-order subdiffusion equation. Liu et al. [45] developed an implicit meshless approach based on the radial basis function for the numerical simulation of a two-dimensional subdiffusion problem. Chen and Liu [18] considered an implicit difference scheme for a three-dimensional fractional advection-diffusion equation, and a Richardson extrapolation was applied to improve the accuracy. The complexity of the fractional differential equations comes from the involving fractional derivatives that are nonlocal and have the character of history dependence and universal mutuality. This means that the computations would be costly if the implicit schemes were applied, especially for solving multidimensional problems [43, 44]. Some researchers have explored some techniques for reducing this cost. These techniques include the adaptive technique [46] and the matrix transfer technique [47, 48]. It is well known that alternating direction implicit (ADI) methods are unconditionally stable as the traditional implicit methods. On the other hand, they reduce a multidimensional problem to a series of independent one-dimensional problems, and thus the computational complexities and the computational cost can be greatly reduced. Therefore, ADI methods for fractional differential equations have the potential to
References
[1]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
[2]
R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, and P. Paradisi, “Discrete random walk models for space-time fractional diffusion,” Chemical Physics, vol. 284, pp. 521–541, 2002.
[3]
B. I. Henry and S. L. Wearne, “Fractional reaction-diffusion,” Physica A, vol. 276, no. 3-4, pp. 448–455, 2000.
[4]
M. Giona and H. E. Roman, “Fractional diffusion equation for transport phenomenna in random media,” Physica A, vol. 185, pp. 87–97, 1992.
[5]
T. Kosztolowicz, “Subdiffusion in a system with a thick membrane,” Journal of Membrane Science, vol. 320, pp. 492–499, 2008.
[6]
R. Metzler, E. Barkai, and J. Klafter, “Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach,” Physical Review Letters, vol. 82, pp. 3563–3567, 1999.
[7]
R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, p. 77, 2000.
[8]
R. Metzler and J. Klafter, “Boundary value problems for fractional diffusion equations,” Physica A, vol. 278, no. 1-2, pp. 107–125, 2000.
[9]
M. Li, “Approximating ideal filters by systems of fractional order,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 365054, 6 pages, 2012.
[10]
M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011.
[11]
M. Li and W. Zhao, “Essay on fractional Riemann-Liouville integral operator versus Mikusinski’s,” Mathematical Problems in Engineering, vol. 2013, Article ID 635412, 3 pages, 2013.
[12]
M. Li and W. Zhao, “Solving the Abel's type integral equation with the Mikusinski's operator of fractional order,” Advances in Mathematical Physics, vol. 2013, Article ID 806984, 4 pages, 2013.
[13]
J. A. Tenreiro MacHado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010.
[14]
H. Sun, Y. Chen, and W. Chen, “Random-order fractional differential equation models,” Signal Processing, vol. 91, no. 3, pp. 525–530, 2011.
[15]
C. Cattani, “Fractional calculus and Shannon wavelet,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 26 pages, 2012.
[16]
C. Cattani, G. Pierro, and G. Altieri, “Entropy and multifractality for the myeloma multiple TET 2 gene,” Mathematical Problems in Engineering, vol. 2012, Article ID 193761, 14 pages, 2012.
[17]
B. Baeumer, M. Kovács, and M. M. Meerschaert, “Numerical solutions for fractional reaction-diffusion equations,” Computers & Mathematics with Applications, vol. 55, no. 10, pp. 2212–2226, 2008.
[18]
C.-M. Chen and F. Liu, “A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation,” Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 219–236, 2009.
[19]
C.-M. Chen, F. Liu, I. Turner, and V. Anh, “A Fourier method for the fractional diffusion equation describing sub-diffusion,” Journal of Computational Physics, vol. 227, no. 2, pp. 886–897, 2007.
[20]
C.-M. Chen, F. Liu, V. Anh, and I. Turner, “Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation,” SIAM Journal on Scientific Computing, vol. 32, no. 4, pp. 1740–1760, 2010.
[21]
C.-m. Chen, F. Liu, and K. Burrage, “Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 754–769, 2008.
[22]
A. Mohebbi, M. Abbaszadeh, and M. Dehghan, “A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term,” Journal of Computational Physics, vol. 240, pp. 36–48, 2013.
[23]
G.-H. Gao, Z.-Z. Sun, and Y.-N. Zhang, “A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions,” Journal of Computational Physics, vol. 231, no. 7, pp. 2865–2879, 2012.
[24]
T. A. M. Langlands and B. I. Henry, “The accuracy and stability of an implicit solution method for the fractional diffusion equation,” Journal of Computational Physics, vol. 205, no. 2, pp. 719–736, 2005.
[25]
Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,” Journal of Computational Physics, vol. 225, no. 2, pp. 1533–1552, 2007.
[26]
F. Liu, C. Yang, and K. Burrage, “Numerical method and analytic technique of the modified anomalous subdiffusion equation with a nonlinear source term,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 160–176, 2009.
[27]
F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, “Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 12–20, 2007.
[28]
C.-M. Chen, F. Liu, V. Anh, and I. Turner, “Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation,” Mathematics of Computation, vol. 81, no. 277, pp. 345–366, 2012.
[29]
Q. Liu, F. Liu, I. Turner, and V. Anh, “Numerical simulation for the 3D seepage flow with fractional derivatives in porous media,” IMA Journal of Applied Mathematics, vol. 74, no. 2, pp. 201–229, 2009.
[30]
C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler, “A second-order accurate numerical approximation for the fractional diffusion equation,” Journal of Computational Physics, vol. 213, no. 1, pp. 205–213, 2006.
[31]
S. B. Yuste, “Weighted average finite difference methods for fractional diffusion equations,” Journal of Computational Physics, vol. 216, no. 1, pp. 264–274, 2006.
[32]
S. B. Yuste and L. Acedo, “An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 1862–1874, 2005.
[33]
P. Zhuang, F. Liu, V. Anh, and I. Turner, “New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation,” SIAM Journal on Numerical Analysis, vol. 46, no. 2, pp. 1079–1095, 2008.
[34]
P. Zhuang, F. Liu, V. Anh, and I. Turner, “Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 1760–1781, 2009.
[35]
J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Physics Reports, vol. 195, no. 4-5, pp. 127–293, 1990.
[36]
P. Zhuang, F. Liu, V. Anh, and I. Turner, “Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process,” IMA Journal of Applied Mathematics, vol. 74, no. 5, pp. 645–667, 2009.
[37]
Y. T. Gu, P. Zhuang, and F. Liu, “An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation,” Computer Modeling in Engineering & Sciences, vol. 56, no. 3, pp. 303–333, 2010.
[38]
M. Cui, “Compact finite difference method for the fractional diffusion equation,” Journal of Computational Physics, vol. 228, no. 20, pp. 7792–7804, 2009.
[39]
Y.-N. Zhang, Z.-Z. Sun, and H.-W. Wu, “Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation,” SIAM Journal on Numerical Analysis, vol. 49, no. 6, pp. 2302–2322, 2011.
[40]
G.-h. Gao and Z.-z. Sun, “A compact finite difference scheme for the fractional sub-diffusion equations,” Journal of Computational Physics, vol. 230, no. 3, pp. 586–595, 2011.
[41]
X. Li and C. Xu, “A space-time spectral method for the time fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 2108–2131, 2009.
[42]
X. Zhao and Z.-z. Sun, “A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions,” Journal of Computational Physics, vol. 230, no. 15, pp. 6061–6074, 2011.
[43]
C.-M. Chen, F. Liu, I. Turner, and V. Anh, “Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation,” Numerical Algorithms, vol. 54, no. 1, pp. 1–21, 2010.
[44]
P. Zhuang and F. Liu, “Finite difference approximation for two-dimensional time fractional diffusion equation,” Journal of Algorithms & Computational Technology, vol. 1, no. 1, pp. 1–15, 2007.
[45]
Q. Liu, Y. T. Gu, P. Zhuang, F. Liu, and Y. F. Nie, “An implicit RBF meshless approach for time fractional diffusion equations,” Computational Mechanics, vol. 48, no. 1, pp. 1–12, 2011.
[46]
H. Brunner, L. Ling, and M. Yamamoto, “Numerical simulations of 2D fractional subdiffusion problems,” Journal of Computational Physics, vol. 229, no. 18, pp. 6613–6622, 2010.
[47]
Q. Yang, T. Moroney, K. Burrage, I. Turner, and F. Liu, “Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions,” ANZIAM Journal. Electronic Supplement, vol. 52, pp. C395–C409, 2010.
[48]
Q. Yang, I. Turner, F. Liu, and M. Ili？, “Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions,” SIAM Journal on Scientific Computing, vol. 33, no. 3, pp. 1159–1180, 2011.
[49]
Q. X. Liu and F. W. Liu, “Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives,” Mathematica Numerica Sinica, vol. 31, no. 2, pp. 179–194, 2009.
[50]
S. Chen and F. Liu, “ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation,” Journal of Applied Mathematics and Computing, vol. 26, no. 1-2, pp. 295–311, 2008.
[51]
M. M. Meerschaert, H.-P. Scheffler, and C. Tadjeran, “Finite difference methods for two-dimensional fractional dispersion equation,” Journal of Computational Physics, vol. 211, no. 1, pp. 249–261, 2006.
[52]
H. Wang and K. Wang, “An alternating-direction finite difference method for two-dimensional fractional diffusion equations,” Journal of Computational Physics, vol. 230, no. 21, pp. 7830–7839, 2011.
[53]
C. Tadjeran and M. M. Meerschaert, “A second-order accurate numerical method for the two-dimensional fractional diffusion equation,” Journal of Computational Physics, vol. 220, no. 2, pp. 813–823, 2007.
[54]
Y.-N. Zhang, Z.-Z. Sun, and X. Zhao, “Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation,” SIAM Journal on Numerical Analysis, vol. 50, no. 3, pp. 1535–1555, 2012.
[55]
M. Cui, “Compact alternating direction implicit method for two-dimensional time fractional diffusion equation,” Journal of Computational Physics, vol. 231, no. 6, pp. 2621–2633, 2012.
[56]
S. Chen, F. Liu, P. Zhuang, and V. Anh, “Finite difference approximations for the fractional Fokker-Planck equation,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 256–273, 2009.
[57]
W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008/09.
[58]
Z.-Z. Sun and X. Wu, “A fully discrete difference scheme for a diffusion-wave system,” Applied Numerical Mathematics, vol. 56, no. 2, pp. 193–209, 2006.
[59]
Y.-N. Zhang and Z.-Z. Sun, “Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation,” Journal of Computational Physics, vol. 230, no. 24, pp. 8713–8728, 2011.
[60]
A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, NY, USA, 2001.
