Fast solution methods for Riesz space fractional diffusion equations with non-separable coefficients

In this paper, efficient preconditioned iterative methods for solving the Riesz space fractional diffusion equations with variable diffusion coefficients are considered. Crank–Nicolson and weighted and shifted Grünwald difference scheme are applied to discretize the problem. For one-dimensional problems, we transform the asymmetric discretized linear system into a symmetric one and solve the resulted linear system by preconditioned conjugate gradient method. The preconditioned conjugate gradient method with a symmetric Toeplitz preconditioner and a sine transform based preconditioner are employed to solve the resulting linear system. Theoretically, we respectively prove that the condition numbers of preconditioned matrices have uniform upper bounds, which is independent of discretization step-sizes and fractional order. For two-dimensional problems, we propose an optional and symmetric splitting iteration method, which can be embedded by conjugate gradient method although the diffusion coefficients are non-separable. We prove that the optional and symmetric splitting iteration method is unconditionally convergent. Numerical results are reported to illustrate the efficiency of the proposed methods.