标题： 一种使用广义数值通量的全离散局部间断伽辽金方法，用于求解分数阶反应扩散方程 显示英文标题

标题： A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

摘要： 具有截断效应的分数阶扩散方程可以被看作是经典分数阶扩散方程的推广，其中截断效应包含在有界域中。 本文专注于设计一种基于广义交替数值通量的高阶全离散局部间断伽辽金（LDG）方法，用于求解具有截断效应的分数阶扩散方程。 从实际角度来看，不同于纯粹交替数值通量的广义交替数值通量具有更广泛的应用范围。 我们首先设计了一种高效的有限差分格式来近似具有截断效应的分数阶导数，然后为具有截断效应的分数阶扩散方程设计了一种全离散LDG方法。 我们证明了该方案是无条件稳定且收敛的，收敛阶为$O(h^{k+1}+\tau^{2-\alpha})$，其中$h, \tau$和$k$分别为空间步长、时间步长和分段多项式的次数。 最后进行了数值实验以展示方法的有效性并验证其准确性。 显示英文摘要

摘要： 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.