Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-22T15:10:04.764Z Has data issue: false hasContentIssue false
  • English
  • Français

A Second-Order Finite Difference Method for Two-Dimensional Fractional Percolation Equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Miscellaneous topics - Partial differential equations Real functions

Published online by Cambridge University Press:  16 March 2016

Boling Guo ,
Qiang Xu  and
Ailing Zhu
Boling Guo
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Qiang Xu*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China
Ailing Zhu
Affiliation:
School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China
*
* Corresponding author. Email addresses:gbl@iapcm.ac.cn (B. Guo), xuqiangsdu@gmail.com (Q. Xu), zhuailing88@126.com (A. Zhu)
Get access

Abstract

A finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Consistency, stability and convergence of the method are established. Numerical experiments illustrating the effectiveness of the theoretical analysis are provided.

Keywords

Fractional percolation equationsecond-order finite difference methodCrank-Nicolson schemeRichardson extrapolationstability analysis

MSC classification

Secondary: 65M06: Finite difference methods 26A33: Fractional derivatives and integrals 35R11: Fractional partial differential equations
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 733 - 757
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Basu, T.S., Wang, H., A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Model., 9 (2012), 658–666.Google Scholar
[2]Burrage, K., Hale, N., Kay, D., An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34 (4) (2012), A2145–A2172.Google Scholar
[3]Chen, C.-M., Liu, F., Burrage, K., Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation, Appl. Math. Comput., 198 (2008), 754–769.Google Scholar
[4]Chen, S., Liu, F., Burrage, K., Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media, Comput.Math. Appl., (2013), doi: 10.1016/j.camwa.2013.01.023.Google Scholar
[5]Chen, S., Liu, F., Turner, I., Anh, V., A novel implicit finite difference methods for the one dimensional fractional percolation equation, Numer. Algor., 56 (2011), 517–535.Google Scholar
[6]Chen, S., Liu, F., Turner, I., Anh, V., An implicit numerical method for the two-dimensional fractional percolation equation, Appl. Math. Comput., 219 (2013), 4322–4331.Google Scholar
[7]Chou, H., Lee, B., Chen, C., The transient infiltration process for seepage flow from cracks, Advances in Subsurface Flow and Transport: Eastern and Western Approaches III, 2006.Google Scholar
[8]Ervin, V.J., Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Part. Different. Equat., 22 (2005), 558–576.Google Scholar
[9]V.J. Ervin, , J.P. Roop, , Variational solution of fractional advection dispersion equations on bounded domains in ℝd, Numer. Methods Part. Different. Equat., 23 (2007), 256–281.CrossRefGoogle Scholar
[10]Ervin, V.J., Heuer, N., Roop, J.P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2) (2007), 572–591.Google Scholar
[11]Gu, Y.T., Zhuang, P., Liu, Q., An advanced meshless method for time fractional diffusion equation, Int. J. Comput. Methods, 8 (2011), 653–665.Google Scholar
[12]He, J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng., 167 (1998), 57–68.CrossRefGoogle Scholar
[13]Huang, J., Nie, N., Tang, Y., A second order finite difference-spectral method for space fractional diffusion equations, Science China Mathematics, Nov (2013), 1-15.Google Scholar
[14]Isaacson, E., Keller, H.B., Analysis of Numerical Methods, Wiley, New York, 1966.Google Scholar
[15]Li, X., Xu, C., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016–1051.Google Scholar
[16]Li, C., Zhao, Z., Chen, Y., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (3) (2011), 855–875.CrossRefGoogle Scholar
[17]Lin, Y., Li, X., Xu, C., Finite dfiference/specrtal approximations for the fractional cable equation, Math. Comp., 80 (2011), 1369–1396.Google Scholar
[18]Lin, F.-R., Yang, S.-W., Jin, X.-Q., Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014), 109–117.CrossRefGoogle Scholar
[19]Liu, F., Chen, S., Turner, I., Burrage, K., Anh, V., Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term, Cent. Eur. J. Phys., 11 (10) (2013), 1221–1232.Google Scholar
[20]Liu, Q., Gu, Y.T., Zhuang, P., Liu, F., Nie, Y.F., An implicit RBF meshless approach for time fractional diffusion equations, Comput. Mech., 48 (2011), 1–12.Google Scholar
[21]Liu, Q., Liu, F., Turner, I., Anh, V., Numerical simulation for the 3D seepage flow with fractional derivatives in porous media, IMA J. Appl. Math., 74 (2009), 201–229.Google Scholar
[22]Liu, F., Turner, I., Anh, V., Yang, Q., Burrage, K., A numerical method for the fractional FitzhughCNagumo monodomain model, ANZIAM J., 54 (2013), 608–629.CrossRefGoogle Scholar
[23]Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K., A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain, J. Comput. Phys., 293 (2015), 252–263.Google Scholar
[24]Luo, Z.-J., Zhang, Y.-Y., Wu, Y.-X., Finite element numerical simulation of three-dimensional seepage control for deep foundation pit dewatering, J. Hydrodyn., Ser. B, 20 (5) (2008), 596–602.CrossRefGoogle Scholar
[25]Meerschaert, M.M., Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77.Google Scholar
[26]Meerschaert, M.M., Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), 80–90.Google Scholar
[27]Meerschaert, M.M., Scheffler, H.P., Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249–261.Google Scholar
[28]Miller, K., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Google Scholar
[29]Momani, S., Rqayiq, A.A., Baleanu, D., A nonstandard finite difference scheme for two-sided space-fractional partial differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (4) (2012), 1250079 1-5.Google Scholar
[30]Ochoa-Tapia, J., Valdes-Parada, F., alvares-Ramirez, J., A fractional-order Darcy's law, Phy. A, 374 (1) (2007), 1–14.Google Scholar
[31]Petford, N., Koenders, M.A., Seepage flow and consolidation in a deforming porous medium, Geophys. Res. Abstracts, 5 (2003), 13329.Google Scholar
[32]Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
[33]Richtmyer, R.D., Morton, K.W., Difference Methods for Initial-Value Problems, Krieger Publishing, Malabar, FL, 1994.Google Scholar
[34]Rushton, K.R., Redshaw, S.C., Seepage and groundwater flow, Brisbane, Australia: Wiley-Interscience Publication, 1979.Google Scholar
[35]Samko, S., Kilbas, A., Marichev, O., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993.Google Scholar
[36]Song, J., Yu, Q., Liu, F., Turner, I., A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation, Numer. Algorithms, 66 (4) (2014), 911–932.Google Scholar
[37]Sweilam, N., Khader, M., Nagy, A., Numerical solution of two-sided space-fractional wave equation using finite difference method, J. Comput. Appl. Math., 235 (2011), 2832–2841.CrossRefGoogle Scholar
[38]Tadjeran, C., Meerschaert, M.M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys., 220 (2007), 813–823.Google Scholar
[39]Tadjeran, C., Meerschaert, M.M., Scheffler, H.P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), 205–213.Google Scholar
[40]Thusyanthan, N.I., Madabhushi, S.P.G., Scaling of seepage flow velocity in centrifuge models, CUED/D-SOILS/TR326, 2003.Google Scholar
[41]Tian, W., Zhou, H., Deng, W., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., in press (arXiv:1201.5949).Google Scholar
[42]Wang, H., Du, N.A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49–57.Google Scholar
[43]Wang, H., Wang, K., Sircar, T., A direct finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095–8104.CrossRefGoogle Scholar
[44]Yang, Q., Moroney, T., Burrage, K., Turner, I., Liu, F., Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions, Aust. New Zealand Ind. Appl. Math. J., 52 (2011), C395–C409.Google Scholar
[45]Yang, Q., Turner, I., Moroney, T., Liu, F., A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction-diffusion equations, Applied Mathematical Modelling, 38 (15-16) (2014), 3755–3762.Google Scholar
[46]Yu, Q., Liu, F., Turner, I., Burrage, K., A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D, Appl. Math. Comput., 219 (8) (2012), 4082–4095.Google Scholar
[47]Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I., Anh, V., A Crank-Nicolson ADI spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52 (6) (2014), 2599–2622.Google Scholar
[48]Zhang, X., Crawford, J.W., Deeks, L.K., Stutter, M.I., Bengough, A.G., Young, I.M., A mass balance based numerical method for the fractional advection-dispersion equation: Theory and application, Water Resour. Res., 41 (2005), 1–10.Google Scholar
[49]Zhang, Y., Sun, Z., Zhao, X., Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal., 50 (3) (2012), 1535–1555.Google Scholar
[50]Zhou, H., Tian, W., Deng, W., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput., 56 (1) (2013), 45–66.CrossRefGoogle Scholar
[51]Zhuang, P., Liu, F., Anh, V., Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (3) (2009), 1760–1781.Google Scholar