A separable preconditioner for time-space fractional Caputo-Riesz diffusion equations

In this paper, we study linear systems arising from time-space fractional Caputo-Riesz diffusion equations with time-dependent diffusion coefficients. The coefficient matrix is a summation of a block-lower-triangular-Toeplitz matrix (temporal component) and a block-diagonal-with-diagonal-times-Toeplitz-block matrix (spatial component). The main aim of this paper is to propose separable preconditioners for solving these linear systems, where a block ε-circulant preconditioner is used for the temporal component, while a block diagonal approximation is used for the spatial variable. The resulting preconditioner can be block-diagonalized in the temporal domain. Furthermore, the fast solvers can be employed to solve smaller linear systems in the spatial domain. Theoretically, we show that if the diffusion coefficient (temporal-dependent or spatial-dependent only) function is smooth enough, the singular values of the preconditioned matrix are bounded independent of discretization parameters. Numerical examples are tested to show the performance of proposed preconditioner.