Preconditioning techniques of all-at-once systems for multi-term time-fractional diffusion equations

In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner.