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 satisfies . We combine the quadratic spline collocation (QSC) method and the formula to propose a QSC- scheme. It can be proved that, the QSC- scheme is unconditionally stable and convergent with , where , Δ and Δ are the temporal and spatial step sizes, respectively. With some restrictions on , the QSC- scheme has second order convergence in time 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- scheme, which has the same unconditionally stability and convergence orders. In addition, a fast implementation for the ADI-QSC- 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.

中文翻译：

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基于二次样条配置法和平均L1格式的变阶时间分数动/不动扩散方程的高阶数值方法


在本文中，我们考虑二维空间域中的变阶时间分数移动/不动扩散（TF-MID）方程，其中分数阶满足 。我们结合二次样条配置（QSC）方法和公式提出了 QSC 方案。可以证明，QSC-方案无条件稳定且收敛于 ，其中 、Δ 和 Δ 分别是时间和空间步长。在 的一些限制下，QSC-方案即使在均匀网格上也具有时间二阶收敛性，且对方程的解没有任何限制。我们进一步构建了一种新颖的交替方向隐式（ADI）框架来开发ADI-QSC-方案，该方案具有相同的无条件稳定性和收敛阶数。此外，还提出了一种基于指数和逼近（ESA）技术的 ADI-QSC 方案的快速实现。此外，我们还引入了最优 QSC 方法，将空间收敛性提高到四阶。附有数值实验来支持理论分析，并证明所提出方案的有效性。