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In this paper, we develop a fast numerical method for solving thetime-dependent Riesz space fractional diffusion equations with a nonlinearsource term in the convex domain. An implicit finite difference method isemployed to discretize the Riesz space fractional diffusion equations with apenalty term in a rectangular region by the volume-penalization approach. Thestability and the convergence of the proposed method are studied. As thecoefficient matrix is with the Toeplitz-like structure, the generalized minimumresidual method with a preconditioner based on the sine transform is exploitedto solve the discretized linear system, where the preconditioner is constructedin view of the combination of two approximate inverse ${\tau}$ matrices, whichcan be diagonalized by the sine transform. The spectrum of the preconditionedmatrix is also investigated. Numerical experiments are carried out todemonstrate the efficiency of the proposed method.

A preconditioner based on sine transform for two-dimensional Riesz space factional diffusion equations in convex domains