The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order $ O(h^{k+1}+\tau^{2-\alpha}) $, where $ h, \tau $ and $ k $ are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.

中文翻译：

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具有广义数值通量的全离散局部不连续Galerkin方法，用于求解回火的分数反应扩散方程

回火的分数扩散方程可以看作是经典的分数扩散方程的推广，因为截断效应包括在有界域中。本文针对回火分数扩散方程，基于广义交替数值通量，设计高阶完全离散局部不连续伽勒金（LDG）方法。从实践的角度来看，不同于纯交变数值通量的广义交变数值通量具有广泛的应用范围。我们首先设计一个有效的有限差分方案来近似回火的分数阶导数，然后设计一个用于回火的分数扩散方程的完全离散的LDG方法。我们证明该方案是无条件稳定的，并且收敛于阶$ O（h ^ {k + 1} + \ tau ^ {2- \ alpha}）$，其中$ h，\ tau $和$ k $是阶跃空间的大小，时间和分段多项式的阶数。最后通过数值实验证明了该方法的有效性并验证了该方法的准确性。