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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 (Δ t 2 + Δ x 4 + Δ y 4) in the discrete L 2 -norm, where Δ t , Δ x and Δ y 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 (N x N y log ⁡ N x N y) , 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 τ -preconditioner for solving the discretized linear system has a convergence rate independent of discretization stepsizes. [ABSTRACT FROM AUTHOR]

We consider the solution of differential equations with multidelays by using boundary value methods (BVMs). These methods require the solution of some nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed to solve these linear systems. If an Ak1,k2-stable BVM is used, we show that our preconditioner is invertible and the spectrum of the preconditioned matrix is clustered. It follows that when the GMRES method is applied to solving the preconditioned systems, the method would converge fast. Numerical results are given to show the effectiveness of our methods. [Copyright &y& Elsevier]