In this paper, we consider the variable-order time fractional mobile/immobile diffusion (TF-MID) equation in two-dimensional spatial domain, where the fractional order $\alpha(t)$ satisfies $0<\alpha_{*}\leq \alpha(t)\leq \alpha^{*}<1$. We combine the quadratic spline collocation (QSC) method and the $L1^+$ formula to propose a QSC-$L1^+$ scheme. It can be proved that, the QSC-$L1^+$ scheme is unconditionally stable and convergent with $\mathcal{O}(\tau^{\min{\{3-\alpha^*-\alpha(0),2\}}} + \Delta x^{2}+\Delta y^{2})$, where $\tau$, $\Delta x$ and $\Delta y$ are the temporal and spatial step sizes, respectively. With some proper assumptions on $\alpha(t)$, the QSC-$L1^+$ scheme has second temporal convergence order even on the uniform mesh, without any restrictions on the solution of the equation. We further construct a novel alternating direction implicit (ADI) framework to develop an ADI-QSC-$L1^+$ scheme, which has the same unconditionally stability and convergence orders. In addition, a fast implementation for the ADI-QSC-$L1^+$ scheme based on the exponential-sum-approximation (ESA) technique is proposed. Moreover, we also introduce the optimal QSC method to improve the spatial convergence to fourth-order. Numerical experiments are attached to support the theoretical analysis, and to demonstrate the effectiveness of the proposed schemes.

翻译：本文考虑二维空间域上的变阶时间分数阶移动/固定扩散（TF-MID）方程，其中分数阶α(t)满足0<α_*≤α(t)≤α^*<1。我们将二次样条配置方法与L1^+公式结合，提出QSC-L1^+格式。理论证明，该格式无条件稳定且收敛阶为O(τ^{min{3-α^*-α(0),2}}+Δx^2+Δy^2)，其中τ、Δx和Δy分别为时间和空间步长。在α(t)满足适当假设条件下，即使采用均匀网格，QSC-L1^+格式亦可实现二阶时间收敛精度，且无需对方程的解施加任何限制。我们进一步构建新型交替方向隐式框架，发展出具有相同无条件稳定性和收敛阶的ADI-QSC-L1^+格式。此外，基于指数和逼近技术，提出ADI-QSC-L1^+格式的快速实现方案。同时引入最优二次样条配置方法将空间收敛阶提升至四阶。数值实验验证了理论分析结果，并证明了所提方法的有效性。