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Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

Published online by Cambridge University Press:  06 March 2015

Hongxing Rui  and
Jian Huang
Hongxing Rui*
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
Jian Huang
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, China
*
*Corresponding author. Email addresses: [email protected] (H.-X. Rui), [email protected] (J. Huang)
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Abstract

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete l2 norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

Keywords

65M0665M1265M15Finite difference schemefractional diffusion equationuniformly stableexplicitly solvable methodasymmetric techniqueerror estimate
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 1 , February 2015 , pp. 29 - 47
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Baeumer, B., Meerschaert, M.M., Benson, D.A. and Wheatcraft, S.W., Subordinated advection-dispersion equation for contaminant transport, Water Resources Res. 37, 1543–1550 (2001).Google Scholar
[2]Barkai, E., Metzler, R. and Klafter, J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E 61, 132–138 (2000).CrossRefGoogle Scholar
[3]Benson, D., Wheatcraft, S. and Meerschaert, M.M., The fractional-order governing equation of Levy motions, Water Resources Res. 36, 1413–1424 (2000).Google Scholar
[4]Blumen, A., Zumofen, G. and Klafter, J., Transport aspect in anomalous diffusion: Levy walks, Phys. Rev. A 40, 3964 (1989).Google Scholar
[5]Chaves, A., Fractional diffusion equation to describe Levy flights, Phys. Lett. A 239 13–16, (1998).Google Scholar
[6]Chen, M. and Deng, W., Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators, Commun. Comp. Phys. 16, 516–540 (2014).Google Scholar
[7]Concezzi, M. and Spigler, R., Numerical solution of two-dimensional FDE by a high-order ADI method, Comm. Appl. Ind. Math. 3, No. 2, e-421 (2012).Google Scholar
[8]Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comp. Phys. 228, 7792–7804 (2009).Google Scholar
[9]Deng, Z., Singh, V.P. and Bengtsson, L., Numerical solution of fractional advection-dispersion equation, J. Hydraulic Eng. 130, 422–431 (2004).CrossRefGoogle Scholar
[10]Ervin, V.J., Heuer, N. and Roop, J.P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Num. Anal. 45, 572–591 (2007).Google Scholar
[11]Ervin, V.J. and Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Num. Meth. Part. Diff. Eqns. 22, 558–576 (2005).Google Scholar
[12]Evans, D.J. and Abdullah, A.R.B., Group explicit methods for parabolic equations, Int. J. Comp. Math. 14, 73–105 (1983).Google Scholar
[13]Evans, D.J. and Abdullah, A.R.B., A new explicit methods for the diffusion-convection equation, Comp. & Math. with App. 11, 145–154 (1985).Google Scholar
[14]Ji, X. and Tang, H., High-Order accurate Runge-Kutta (local) discontinuous Galerkin methods for one- and two-dimensional fractional diffusion equations, Num. Math. Theor. Meth. App. 5, 333–358 (2012).Google Scholar
[15]Liu, F., Anh, V. and Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 166, 209–219 (2004).CrossRefGoogle Scholar
[16]Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).CrossRefGoogle Scholar
[17]Isaacson, E. and Keller, H.B., Analysis of Numerical Methods, Wiley, New York (1966).Google Scholar
[18]Kirchner, J. W., Feng, X. and Neal, C., Fractal stream chemistry and its implications for contaminant transport in catchments, Nature 403, 524–526 (2000).Google Scholar
[19]Langlands, T.A.M. and Henry, B.I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys. 205, 719–736 (2005).Google Scholar
[20]Li, X. and Xu, C., The existence and uniqueness of the week solution of the space-time fractional diffusion equation and a spectral method approximation, Comm. Comp. Phys. 8, 1016–1051 (2010).Google Scholar
[21]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comp. Phys. 225, 1533–1552 (2007).Google Scholar
[22]Lin, R., Liu, F., Anh, V. and Turner, I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comp. 212, 435–445 (2009).Google Scholar
[23]Liu, F. and Anh, V., Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 166, 209–219 (2004).Google Scholar
[24]Lynch, V.E., Carreras, B.A., Del-Castillo-Negrete, D., Ferreria-Mejias, K.M. and Hicks, H.R., Numerical methods for the solution of partial differential equations of fractional order, J. Comp. Phys. 192, 406–421 (2003).CrossRefGoogle Scholar
[25]Meerschaert, M.M., Scheffler, H.P. and Tadjeran, C., Finite difference methods for the two-dimensional fractional dispersion equation, J. Comp. Phys. 211, 249–261 (2006).Google Scholar
[26]Meerschaert, M.M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equation, J. Comp. Appl. Math. 172, 65–77 (2004).Google Scholar
[27]Meerschaert, M. and Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56, 80–90 (2006).Google Scholar
[28]Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).Google Scholar
[29]Raberto, M., Scalas, E. and Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314, 749–755 (2001).Google Scholar
[30]Sabatelli, L., Keating, S., Dudley, J. and Richmond, P., Waiting time distributions in financial markets, Eur. Phys. J. B 27, 273–275 (2002).Google Scholar
[31]Saulyev, V.K., Integration of Equations of Parabolic Type by the Method of Nets, Pergamon Press, New York (1964).Google Scholar
[32]Scalas, E., Gorenflo, R. and Mainardi, F., Fractional calculus and continuous-time finance, Phys. A 284, 376–384 (2000).CrossRefGoogle Scholar
[33]Schumer, R., Benson, D.A., Meerschaert, M.M. and Baeumer, B., Multiscaling fractional advection-dispersion equations and their solutions, Water Resources Res. 39,1022–1032 (2003).Google Scholar
[34]Schumer, R., Benson, D.A., Meerschaert, M.M. and Wheatcraft, S.W., Eulerian derivation of the fractional advection-dispersion equation, J. Contaminant Hydrol. 48, 69–88 (2001).Google Scholar
[35]Sokolov, I.M., Klafter, J. and Blumen, A., Fractional kinetics, Physics Today 55, 28–53 (2002).Google Scholar
[36]Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. Comp. Phys. 228, 4038–4054 (2009).Google Scholar
[37]Su, L., Wang, W. and Yang, Z., Finite difference approximations for the fractional advection-diffusion equation, Phys. Lett. A 373, 4405–4408 (2009).Google Scholar
[38]Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comp. Phys. 213, 205–213 (2006).Google Scholar
[39]Xu, M. and Tan, W., Theoretical analysis of the velocity field and vortex sheet of generalized second order fluid with fractional anomalous diffusion, Sci. China Ser. A: Math. 44, 1387–1399 (2001).Google Scholar
[40]Ye, X. and Xu, C., Spectral optimization methods for the time fractional diffusion inverse problem, Num. Math. Theor. Meth. Appl. 6, 499–519 (2013).Google Scholar
[41]Zaslavsky, G., Fractional kinetic equation for Hamiltonian Chaotic advection, tracer dynamics and turbulent dispersion, Physica D 76, 110–122 (1994).Google Scholar
[42]Zhang, B., Alternating difference block methods and their difference graphs, Science China Technological Sc. 41, 482–487(1998).Google Scholar
[43]Zhang, B., Difference graphs of block ADI method, SIAM J. Num. Anal. 38, 742–752 (2000).Google Scholar