A second-order accurate numerical approximation for the fractional diffusion equation

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we examine a practical numerical method which is second-order accurate in time and in space to solve a class of initial-boundary value fractional diffusive equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional Crank-Nicholson method based on the shifted Grunwald formula is unconditionally stable. A numerical example is presented and compared with the exact analytical solution for its order of convergence.

[1]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[2]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[3]  Vijay P. Singh,et al.  Numerical Solution of Fractional Advection-Dispersion Equation , 2004 .

[4]  G. Fix,et al.  Least squares finite-element solution of a fractional order two-point boundary value problem , 2004 .

[5]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[6]  Emilia Bazhlekova,et al.  Fractional evolution equations in Banach spaces , 2001 .

[7]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[8]  H. R. Hicks,et al.  Numerical methods for the solution of partial difierential equations of fractional order , 2003 .

[9]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[10]  Aleksei V. Chechkin,et al.  Lévy Flights in a Steep Potential Well , 2003, cond-mat/0306601.

[11]  I. Podlubny.  Fractional differential equations , 1998 .

[12]  D. Benson,et al.  Eulerian derivation of the fractional advection-dispersion equation. , 2001, Journal of contaminant hydrology.

[13]  H. Keller,et al.  Analysis of Numerical Methods , 1967 .

[14]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[15]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[16]  Hans-Peter Scheffler,et al.  Governing equations and solutions of anomalous random walk limits. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Vu Kim Tuan,et al.  Extrapolation to the Limit for Numerical Fractional Differentiation , 1995 .