An efficient preconditioner for linear systems arising from high-order accurate schemes of time fractional diffusion equations

In this paper, we study preconditioners for all-at-once systems arising from the discretization of time-fractional sub-diffusion equations. Due to the use of high-order accurate formulas in time fractional derivative, the coefficient matrix does not have a Toeplitz structure. We reconstructed the coefficient matrix so that the all-at-once system has a non-symmetric Toeplitz-like structure. Based on the non-symmetric Toplitz-like structure of the new system, we designed a preconditioner that can be quickly diagonalized by discrete sine transform and fast Fourier transform techniques. we show that the spectrum of the preconditioned matrix are clustered around 1. Also, we verified the effectiveness of the proposed preconditioner by numerical experiments.