Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property
Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property
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DOI：
10.1137/20m1336503
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发表时间：
2020-05
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影响因子：
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通讯作者：
Zhichao Peng;Yingda Cheng;Jing-Mei Qiu;Fengyan Li
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文献类型：
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作者：
Zhichao Peng;Yingda Cheng;Jing-Mei Qiu;Fengyan Li
在我们最近的工作[22]中，设计了一系列高阶渐进保留（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电报方程。扩散缩放。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。傅立叶分析和数值实验都表明，该方法是无条件稳定的扩散制度时，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顺序的空间，与一般的材料特性的模型。我们的分析是第一个同时确认无条件稳定时，$\varepsilon\ll1 $和一致稳定属性关于$\varepsilon\ll1 $。为了捕捉无条件稳定性，一个新的离散能量被引入更好地探索在不同的制度中的散射项的贡献。权函数的一般形式，引入获得的无条件稳定性$\varepsilon\ll1$，也被认为是第一次在这样的稳定性分析。在一致稳定性的基础上，通过严格的渐近分析证明了该系统的AP性质。
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.