A fast second-order accurate method for a two-sided space-fractional diffusion equation with variable coefficients

In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficients on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann–Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second-order accuracy of O(τ²+h²)O(τ²+h²). Furthermore, we develop a fast accurate iterative method for the Crank–Nicolson scheme, which only requires storage of O(m)O(m) and computational cost of O(mlogm)O(mlogm) while retaining the same accuracy and approximation property as Gauss elimination, where m=1/hm=1/h is the partition number in space direction. Finally, several numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis.