Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling
Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling
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DOI：
10.1016/j.jcp.2020.109485
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发表时间：
2020-08
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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
输运方程出现在许多应用中，如稀薄气体动力学、中子输运和辐射传递。在这项工作中，我们考虑了扩散标度中的一些线性动力学输运方程，并在不连续伽辽金方法框架内设计了高阶渐近保持（AP）方法，其主要目标是在Knudsen数ε≪1时实现扩散区中的无条件稳定性，并在ε= O(1)和ε≪1时实现高阶精度。初始层也被考虑在内。实现这一目标的要素包括：基于微观宏观分解和极限扩散方程的模型重构，空间上的局部不连续伽辽金（LDG）方法，时间上的全局刚性精确隐式-显式（IMEX）龙格-库塔方法，以及处理非精心准备的初始数据的策略。对微宏分解框架内的连续模型进行了形式渐近分析，导出了初始层和初始条件为ε→0渐近一致的内部问题，并对数值格式进行了显示AP性质和理解初始层存在时的数值初始处理。进行了傅里叶稳定性分析，证实了扩散态的无条件稳定性，并给出了ε= 0(1)时动力学态的稳定条件。在重新表述的步骤中，通过增加和减去一个加权扩散项来消除抛物刚度，增强扩散区的数值稳定性。这种想法并不新鲜，但我们的数值稳定性和渐近分析为权重函数的期望性质提供了新的数学理解。最后，通过数值算例验证了所提方法的准确性、稳定性和渐近保持性，以及在初始层存在时所提策略的有效性。
Transport equations arise in many applications such as rarefied gas dynamics, neutron transport, and radiative transfer. In this work, we consider some linear kinetic transport equations in a diffusive scaling and design high order asymptotic preserving (AP) methods within the discontinuous Galerkin method framework, with the main objective to achieve unconditional stability in the diffusive regime when the Knudsen number ε≪ 1, and to achieve high order accuracy when ε= O (1) and when ε≪ 1. Initial layers are also taken into account. The ingredients to accomplish our goal include: model reformulations based on the micro-macro decomposition and the limiting diffusive equation, local discontinuous Galerkin (LDG) methods in space, globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta methods in time, and strategies to handle non-well prepared initial data. Formal asymptotic analysis is carried out for the continuous model within the micro-macro decomposed framework to derive the initial layer as well as the interior problem with an asymptotically consistent initial condition as ε→ 0, and it is also conducted for numerical schemes to show the AP property and to understand the numerical initial treatments in the presence of initial layers. Fourier type stability analysis is performed, and it confirms the unconditional stability in the diffusive regime, and moreover it gives the stability condition in the kinetic regime when ε= O (1). In the reformulation step, a weighted diffusive term is added and subtracted to remove the parabolic stiffness and enhance the numerical stability in the diffusive regime. Such idea is not new, yet our numerical stability and asymptotic analysis provide new mathematical understanding towards the desired properties of the weight function. Finally, numerical examples are presented to demonstrate the accuracy, stability, and asymptotic preserving property of the proposed methods, as well as the effectiveness of the proposed strategies in the presence of the initial layer.