SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1194-A1220, January 2021.
We consider a linear kinetic transport equation under a diffusive scaling that converges to a diffusion equation as the Knudsen number $\varepsilon\rightarrow 0$. In [S. Boscarino, L. Pareschi, and G. Russo, SIAM J. Sci. Comput., 35 (2013), pp. A22--A51; Z. Peng et al., J. Comput. Phys., 415 (2020), 109485], to achieve the asymptotic preserving (AP) property and unconditional stability in the diffusive regime with $\varepsilon\ll 1$, numerical schemes are developed based on an additional reformulation of the even-odd or micro-macro decomposed version of the equation. The key of the reformulation is to add a weighted diffusive term on both sides of one equation in the decomposed system. The choice of the weight function, however, is problem-dependent and ad-hoc, and it can affect the performance of numerical simulations. To avoid issues related to the choice of the weight function and still obtain the AP property and unconditional stability in the diffusive regime, we propose in this paper a new family of AP schemes, termed as IMEX-DG-S schemes, directly solving the micro-macro decomposed system without any further reformulation. The main ingredients of the IMEX-DG-S schemes include globally stiffly accurate implicit-explicit (IMEX) Runge--Kutta temporal discretizations with a new IMEX strategy, discontinuous Galerkin spatial discretizations, discrete ordinate methods for the velocity space, and the application of the Schur complement to the algebraic form of the schemes to control the overall computational cost. The AP property of the schemes is shown formally. With an energy type stability analysis applied to the first order scheme, and Fourier type stability analysis applied to the first to third order schemes, we confirm the uniform stability of the methods with respect to $\varepsilon$ and the unconditional stability in the diffusive regime. A series of numerical examples are presented to demonstrate the performance of the new schemes.

中文翻译：

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基于Schur补码的线性动力学输运方程的渐近守恒IMEX-DG-S格式

SIAM科学计算杂志，第43卷，第2期，第A1194-A1220页，2021年1月。
我们考虑在扩散标度下的线性动力学输运方程，当Knudsen数$ \ varepsilon \ rightarrow 0 $收敛到扩散方程时。在[S. Boscarino，L.Pareschi和G.Russo，SIAM J. Sci。计算（35）（2013），A22--A51页; Z. Peng等人，计算机科学杂志（J. Comput。）。Phys。，415（2020），109485]，以在具有$ \ varepsilon \ ll 1 $的扩散状态下实现渐近保持（AP）性质和无条件稳定性，并基于对奇数的额外重新公式化，开发了数值方案或方程的微宏分解版本。重新制定公式的关键是在分解系统中的一个方程式的两侧添加加权扩散项。但是，权重函数的选择取决于问题且是临时性的，并且会影响数值模拟的性能。为了避免与权函数的选择有关的问题，并且仍然获得扩散状态下的AP性质和无条件稳定性，我们在本文中提出了一个新的AP方案系列，称为IMEX-DG-S方案，直接解决了微观问题。 -宏分解的系统，无需任何进一步的重新构造。IMEX-DG-S方案的主要内容包括采用新的IMEX策略的全局严格精确的隐式显式（IMEX）Runge-Kutta时间离散化，不连续的Galerkin空间离散化，速度空间的离散纵坐标方法以及Schur对方案的代数形式进行了补充，以控制总体计算成本。方案的AP属性已正式显示。通过将能量类型稳定性分析应用于一阶方案，并将傅里叶类型稳定性分析应用于一阶到三阶方案，我们确认了方法在\ varepsilon $上的一致稳定性以及在扩散状态下的无条件稳定性。给出了一系列数值示例，以证明新方案的性能。