稳定性增强的AP IMEX1-LDG方法：基于能量的稳定性和严格AP性质
Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property
摘要：
在我们最近的工作[22]中，设计了一族高阶渐近保持（AP）方法，称为IMEX-LDG方法，用于求解一些线性动力学输运方程，包括板几何中的单群输运方程和扩散标度中的电报方程。当Knudsen数$\varepsilon$趋于零时，极限格式是对极限扩散方程的隐式离散。傅里叶分析和数值实验都表明，当$\varepsilon\ll1$。在本文中，我们开发了一种能量方法来建立数值稳定性的IMEX 1-LDG方法，该方法的子族，是一阶精度的时间和任意阶的空间，具有一般的材料属性的模型。我们的分析是第一个同时确认无条件稳定时，$\varepsilon\ll1$和一致稳定性属性关于$\varepsilon$。为了捕捉无条件稳定性，一个新的离散能量被引入更好地探索在不同的制度中的散射项的贡献。权函数的一般形式，引入获得无条件稳定的$\varepsilon\ll1$，也是第一次考虑在这样的稳定性分析。基于一致稳定性，对该系统进行了严格的渐近分析，证明了该系统的AP性质。
英文摘要：
In our recent work [22], a family of high order asymptotic preserving (AP) methods, termed as IMEX-LDG methods, are designed to solve some linear kinetic transport equations, including the one-group transport equation in slab geometry and the telegraph equation, in a diffusive scaling. As the Knudsen number $\varepsilon$ goes to zero, the limiting schemes are implicit discretizations to the limiting diffusive equation. Both Fourier analysis and numerical experiments imply the methods are unconditionally stable in the diffusive regime when $\varepsilon\ll1$. In this paper, we develop an energy approach to establish the numerical stability of the IMEX1-LDG method, the sub-family of the methods that is first order accurate in time and arbitrary order in space, for the model with general material properties. Our analysis is the first to simultaneously confirm unconditional stability when $\varepsilon\ll1$ and the uniform stability property with respect to $\varepsilon$. To capture the unconditional stability, a novel discrete energy is introduced by better exploring the contribution of the scattering term in different regimes. A general form of the weight function, introduced to obtain the unconditional stability for $\varepsilon\ll1$, is also for the first time considered in such stability analysis. Based on the uniform stability, a rigorous asymptotic analysis is then carried out to show the AP property.
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