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The accuracy and stability of an implicit solution method for the fractional diffusion equation. (English) Zbl 1072.65123
Summary: We have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation. This model equation governs the evolution for the probability density function that describes anomalously diffusing particles. Anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur. The implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement. The accuracy of the scheme is O(Δx 2 ) in the spatial grid size and O(Δt 1+γ ) in the fractional time step, where 01-γ<1 is the order of the fractional derivative and γ=1 is standard diffusion. We have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0<γ1.

MSC:
65M12Stability and convergence of numerical methods (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
35K05Heat equation