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.

研究了由分数阶移动/不移动对流扩散方程产生的全一次系统。首先，采用L 1公式的有限差分法对其进行离散化。所得的隐式方案是时间步进方案，不适用于并行计算。基于此方案，建立了一个适用于并行计算的一次性系统。其次，根据一次系统的块下三角Toeplitz结构，设计了块双对角预处理器和块楼梯预处理器。最后，通过数值例子说明了所提出的预处理器的性能。