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

Authors:
Charles Tadjeran;Mark M. Meerschaert;Hans-Peter Scheffler
Affiliations:
Department of Physics, University of Nevada, Reno, NV 89557, USA;Department of Mathematics and Statistics, University of Otago, Room 232A, Dunedin 9001, New Zealand;Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA
Venue:
Journal of Computational Physics
Year:
2006

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Abstract

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.