Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

In this paper, we consider a space–time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0, 1], and the first-order and second-order space derivatives by the Riemman–Liouville fractional derivatives of order β ∈ (0, 1] and of order γ ∈ (1, 2], respectively. For the space fractional derivatives Dxβu(x,t) and Dxγu(x,t), we adopted the Grünwald formula and the shift Grünwald formula, respectively. We propose an implicit difference method (IDM) and an explicit difference method (EDM) to solve this equation. Stability and convergence of these methods are discussed. Using mathematical induction, we prove that the IDM is unconditionally stable and convergent, but the EDM is conditionally stable and convergent. Numerical results are in good agreement with theoretical analysis.