Skip to main content Accessibility help

Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

  • Hongxing Rui (a1) and Jian Huang (a1)


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.


Corresponding author

*Corresponding author. Email addresses: (H.-X. Rui), (J. Huang)


Hide All
[1]Baeumer, B., Meerschaert, M.M., Benson, D.A. and Wheatcraft, S.W., Subordinated advection-dispersion equation for contaminant transport, Water Resources Res. 37, 15431550 (2001).
[2]Barkai, E., Metzler, R. and Klafter, J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E 61, 132138 (2000).
[3]Benson, D., Wheatcraft, S. and Meerschaert, M.M., The fractional-order governing equation of Levy motions, Water Resources Res. 36, 14131424 (2000).
[4]Blumen, A., Zumofen, G. and Klafter, J., Transport aspect in anomalous diffusion: Levy walks, Phys. Rev. A 40, 3964 (1989).
[5]Chaves, A., Fractional diffusion equation to describe Levy flights, Phys. Lett. A 239 1316, (1998).
[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, 516540 (2014).
[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).
[8]Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comp. Phys. 228, 77927804 (2009).
[9]Deng, Z., Singh, V.P. and Bengtsson, L., Numerical solution of fractional advection-dispersion equation, J. Hydraulic Eng. 130, 422431 (2004).
[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, 572591 (2007).
[11]Ervin, V.J. and Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Num. Meth. Part. Diff. Eqns. 22, 558576 (2005).
[12]Evans, D.J. and Abdullah, A.R.B., Group explicit methods for parabolic equations, Int. J. Comp. Math. 14, 73105 (1983).
[13]Evans, D.J. and Abdullah, A.R.B., A new explicit methods for the diffusion-convection equation, Comp. & Math. with App. 11, 145154 (1985).
[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, 333358 (2012).
[15]Liu, F., Anh, V. and Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 166, 209219 (2004).
[16]Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).
[17]Isaacson, E. and Keller, H.B., Analysis of Numerical Methods, Wiley, New York (1966).
[18]Kirchner, J. W., Feng, X. and Neal, C., Fractal stream chemistry and its implications for contaminant transport in catchments, Nature 403, 524526 (2000).
[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, 719736 (2005).
[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, 10161051 (2010).
[21]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comp. Phys. 225, 15331552 (2007).
[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, 435445 (2009).
[23]Liu, F. and Anh, V., Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 166, 209219 (2004).
[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, 406421 (2003).
[25]Meerschaert, M.M., Scheffler, H.P. and Tadjeran, C., Finite difference methods for the two-dimensional fractional dispersion equation, J. Comp. Phys. 211, 249261 (2006).
[26]Meerschaert, M.M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equation, J. Comp. Appl. Math. 172, 6577 (2004).
[27]Meerschaert, M. and Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56, 8090 (2006).
[28]Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).
[29]Raberto, M., Scalas, E. and Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314, 749755 (2001).
[30]Sabatelli, L., Keating, S., Dudley, J. and Richmond, P., Waiting time distributions in financial markets, Eur. Phys. J. B 27, 273275 (2002).
[31]Saulyev, V.K., Integration of Equations of Parabolic Type by the Method of Nets, Pergamon Press, New York (1964).
[32]Scalas, E., Gorenflo, R. and Mainardi, F., Fractional calculus and continuous-time finance, Phys. A 284, 376384 (2000).
[33]Schumer, R., Benson, D.A., Meerschaert, M.M. and Baeumer, B., Multiscaling fractional advection-dispersion equations and their solutions, Water Resources Res. 39,10221032 (2003).
[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, 6988 (2001).
[35]Sokolov, I.M., Klafter, J. and Blumen, A., Fractional kinetics, Physics Today 55, 2853 (2002).
[36]Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. Comp. Phys. 228, 40384054 (2009).
[37]Su, L., Wang, W. and Yang, Z., Finite difference approximations for the fractional advection-diffusion equation, Phys. Lett. A 373, 44054408 (2009).
[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, 205213 (2006).
[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, 13871399 (2001).
[40]Ye, X. and Xu, C., Spectral optimization methods for the time fractional diffusion inverse problem, Num. Math. Theor. Meth. Appl. 6, 499519 (2013).
[41]Zaslavsky, G., Fractional kinetic equation for Hamiltonian Chaotic advection, tracer dynamics and turbulent dispersion, Physica D 76, 110122 (1994).
[42]Zhang, B., Alternating difference block methods and their difference graphs, Science China Technological Sc. 41, 482487(1998).
[43]Zhang, B., Difference graphs of block ADI method, SIAM J. Num. Anal. 38, 742752 (2000).



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed