A fast second-order scheme for nonlinear Riesz space-fractional diffusion equations

In this paper, the fractional centered difference formula is employed to discretize the Riesz derivative, while the Crank-Nicolson scheme is approximated the time derivative, and the explicit linearized technique is used to deal with nonlinear term, a second-order finite difference method is obtained for the nonlinear Riesz space-fractional diffusion equations, and the resulting system is symmetric positive definite ill-conditioned Toeplitz matrix and then the fast sine transform can be used to reduce the computational cost of the matrix-vector multiplication. The preconditioned conjugate gradient method with a preconditioner based on sine transform is proposed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix falling in an open interval (1/2,3/2) is proved, which can guarantee the linear convergence rate of the proposed methods. By the similar technique, the two-dimension case is also studied. Finally, numerical experiments are carried out to demonstrate that the proposed preconditioner works well.