Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients.

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time and space variables, respectively. With the compact operator acting on, a novel fourth-order finite difference scheme is subsequently constructed. Solvability, stability and convergence of these schemes are theoretically analyzed. For these discretized linear systems, the fast implementation with preconditioners based on sine transform is proposed, which has the computational complexity of O (n log ⁡ n) per iteration and the memory requirement of O (n) , where n represents the total number of the spatial grid nodes. Finally, numerical experiments are performed to illustrate the preciseness and effectiveness of these new techniques. • Two numerical schemes can be proved of stability and high convergence order in space. • The rarely investigated preconditioners can remain the total computational complexity as O (n log ⁡ n). • Our proposed schemes can be flexibly extended to the multi-dimensional problems. [ABSTRACT FROM AUTHOR]