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 CrankNicholson 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 CrankNicholson method based on the shifted Grnwald formula is unconditionally stable. A numerical example is presented and compared with the exact analytical solution for its order of convergence.
cranknicholson method fractional partial differential equation second-order accurate finite difference approximation stability analysis numerical fractional pde numerical algorithm for superdiffusion