This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization. Based on the moving least square theorem and Taylor series, the GFDM introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes on the surface. It inherits the similar properties from the standard FDM and avoids the mesh generation, which is available particularly for high-dimensional irregular discretization nodes. The SD-TSM is a non-uniform temporal discretization method involving the idea of metric, which links the fractional derivative order with the non-uniform discretization strategy. Compared with the traditional time stepping methods, GFDM combined with SD-TSM deals well with the low accuracy in the early period. Numerical investigations are presented to demonstrate the efficiency and accuracy of the proposed GFDM in conjunction with SD-TSM for solving either single or coupled fractional diffusion equations on surfaces.

中文翻译：

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一种用于表面异常扩散的高效局部搭配求解器

本文介绍了一种高效的搭配求解器，即广义有限差分法 (GFDM) 结合最近开发的尺度相关时间步长法 (SD-TSM)，以预测由表面时间分数扩散方程控制的表面上的异常扩散行为. 在建议的求解器中，GFDM 用于空间离散化，SD-TSM 用于时间离散化。GFDM 基​​于移动最小二乘定理和泰勒级数，引入了模板选择算法，从表面上的整个离散化节点中选择某个节点的模板支持。它继承了标准 FDM 的相似特性，避免了网格生成，这尤其适用于高维不规则离散化节点。SD-TSM是一种涉及度量思想的非均匀时间离散化方法，它将分数阶导数与非均匀离散化策略联系起来。与传统的时间步长方法相比，GFDM结合SD-TSM可以很好地解决早期精度低的问题。数值研究展示了所提出的 GFDM 与 SD-TSM 结合用于求解表面上的单个或耦合分数扩散方程的效率和准确性。