Preconditioners for all-at-once system from the fractional mobile/immobile advection–diffusion model

The all-at-once system arising from fractional mobile/immobile advection–diffusion equations is studied. Firstly, the finite difference method with L1 formula is employed to discretize it. The resulting implicit scheme is a time-stepping scheme, which is not suitable for parallel computing. Based on this scheme, an all-at-once system is established, which will be suitable for parallel computing. Secondly, according to the block lower triangular Toeplitz structure of the all-at-once system, both the block bi-diagonal preconditioner and block stair preconditioner are designed. Finally, numerical examples are presented to show the performances of the proposed preconditioners.