The accuracy and stability of an implicit solution method for the fractional diffusion equation

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(Δx2) in the spatial grid size and O(Δt1 + γ) in the fractional time step, where 0 ⩽ 1 − 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 < γ ⩽ 1.