A two-dimensional time-fractional mobile-immobile diffusion problem with the Caputo time-fractional derivative of order \(\alpha \in (0,1)\) is considered. We show that the solution of the problem has a weak singularity at the initial time. Using the averaged L1 formula to approximate the Caputo time-fractional derivative and using a compact finite difference approximation to discretize the space derivatives, we propose a high-order averaged L1-type compact finite difference method on the uniform space-time mesh for the problem. We then base on this method to develop an averaged L1-type compact alternating direction implicit (ADI) finite difference method and a fast sum-of-exponentials compact ADI finite difference method, both of which significantly reduce the storage requirements and the computational costs while maintaining the same global convergence rate. By using the discrete energy analysis technique, we rigorously prove that all methods are unconditionally stable and convergent, and they have the spatial global fourth-order convergence rate and the temporal global convergence rate of order \(\min \{2, 3-2\alpha \}\). For the case of \(\alpha >1/2\), we use the discrete minimum-maximum principle to prove that the temporal second-order convergence rate can also be achieved in positive time. Numerical results confirm the theoretical analysis results and demonstrate the computational efficiency of the methods.

考虑具有\(\alpha \in (0,1)\)阶 Caputo 时间分数导数的二维时间分数移动-不动扩散问题。我们证明问题的解在初始时具有弱奇异性。利用平均L 1 公式逼近Caputo时间分数阶导数，利用紧致有限差分近似离散空间导数，提出了一种均匀时空网格上的高阶平均L 1 型紧致有限差分方法问题。然后，我们基于该方法开发了平均L 1 型紧凑交替方向隐式（ADI）有限差分法和快速指数和紧凑ADI有限差分法，这两种方法都显着降低了存储要求和计算成本同时保持相同的全局收敛速度。通过使用离散能量分析技术，我们严格证明了所有方法都是无条件稳定和收敛的，并且具有空间全局四阶收敛速度和时间全局收敛速度\(\min \{2, 3-2 \α \}\）。对于\(\alpha >1/2\)的情况，我们利用离散最小最大原理证明了在正时间内也能达到时间二阶收敛速度。数值结果证实了理论分析结果并证明了该方法的计算效率。