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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 Real functions Miscellaneous topics - Partial differential equations

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)
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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

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 733 - 757
Copyright
Copyright © Global-Science Press 2016 

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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