论文标题
具有通用数值通量的完全离散的局部不连续的Galerkin方法，用于求解钢化的分数反应扩散方程
A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation
论文作者
论文摘要
回火分数扩散方程可以被认为是经典的分数扩散方程的概括，即截断效应包含在界面域中。本文着重于设计基于回火分数扩散方程的广义交替通量的高阶局部不连续的盖尔金（LDG）方法。从实际的角度来看，与纯粹的交替数值通量不同的广义交替数值通量具有更广泛的应用范围。我们首先设计了有效的有限差方案，以近似回火的分数衍生物，然后使用用于钢化部分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，并且与$ o（h^{k+1}+τ^{2-α}）$的顺序是稳定的，其中$ h，τ$和$ k $分别是空间，时间和分段多项式的时间的步骤。最后，进行数值实验以显示有效性并证明该方法的准确性。
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 $O(h^{k+1}+τ^{2-α})$, where $h, τ$ 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.
