Finite difference/spectral approximations for the time-fractional diffusion equation

In this paper, we consider the numerical resolution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0⩽α⩽1). The main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and Legendre spectral methods in space. Stability and convergence of the method are rigourously established. We prove that the full discretization is unconditionally stable, and the numerical solution converges to the exact one with order O(Δt2-α+N-m), where Δt,N and m are the time step size, polynomial degree, and regularity of the exact solution respectively. Numerical experiments are carried out to support the theoretical claims.