On τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations

In this paper, a τ-preconditioner for a novel fourth-order finite difference scheme of two-dimensional Riesz space-fractional diffusion equations (2D RSFDEs) is considered, in which a fourth-order fractional centered difference operator is adopted for the discretizations of spatial Riesz fractional derivatives, while the Crank-Nicolson method is adopted to discretize the temporal derivative. The scheme is proven to be unconditionally stable and has a convergence rate of O(Δt2+Δx4+Δy4) in the discrete L2-norm, where Δt, Δx and Δy are the temporal and spatial step sizes, respectively. In addition, the preconditioned conjugate gradient (PCG) method with τ-preconditioner is applied to solve the discretized symmetric positive definite linear systems arising from 2D RSFDEs. Theoretically, we show that the τ-preconditioner is invertible by a new technique, and analyze the spectrum of the corresponding preconditioned matrix. Moreover, since the τ-preconditioner can be diagonalized by the discrete sine transform matrix, the total operation cost of the PCG method is O(NxNylog⁡NxNy), where Nx and Ny are the number of spatial unknowns in x- and y-directions. Finally, numerical experiments are performed to verify the convergence orders, and show that the PCG method with the τ-preconditioner for solving the discretized linear system has a convergence rate independent of discretization stepsizes.