On τ-preconditioner for anovel fourth-order difference scheme oftwo-dimensional Riesz space-fractional diffusion equations

In this paper, a tau-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 CrankNicolson method is adopted to discretize the temporal derivative. The scheme is proven to be unconditionally stable and has a convergence rate of circle dot(Delta t(2) + Delta t(4)+ Delta t(4)) in the discrete L-2-norm, where Delta t, Delta x, and Delta y are the temporal and spatial step sizes, respectively. In addition, the preconditioned conjugate gradient (PCG) method with.. iota-preconditioner is applied to solve the discretized symmetric positive definite linear systemsarising from 2D RSFDEs. Theoretically, we show that the iota-preconditioner is invertible by a new technique, and analyze the spectrum of the corresponding preconditioned matrix. Moreover, since the..-preconditioner can be diagonalizedby the discrete sine transform matrix, the total operation cost of the PCG method is circle dot (N-x,N-y log NxNy), where N-x and N-y 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.. -preconditioner for solving the discretized linear system has a convergence rate independent of discretization stepsizes.