论文标题
基于Schur补体的线性动力学方程的IMEX-DG-S方案渐近保存IMEX-DG-S方案
Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on Schur complement
论文作者
论文摘要
我们考虑在扩散缩放下的线性动力学传输方程，该方程会收敛到扩散方程，因为Knudsen Number $ \ varepsilon \ rightarrow0 $。在[3，21]中，为了实现以$ \ varepsilon \ ll 1 $ 1 $的扩散状态中的渐近保留（AP）属性和无条件稳定性，数值方案是基于对等式的均匀ODD或Micro-Macro分解版本的额外重新重新制定的。重新制定的关键是在分解系统中的一个方程式上添加加权扩散术语。但是，重量函数的选择是与问题有关的和临时的，它可能会影响数值模拟的性能。为了避免与重量功能的选择有关的问题，仍然在扩散政权中获得AP特性和无条件的稳定性，我们在本文中提出了一个新的AP方案家族，称为IMEX-DG-S方案，直接求解微型麦克罗分解的系统，而无需进一步重新制定。 IMEX-DG-S方案的主要成分包括具有新的IMEX策略，不连续的Galerkin（DG）空间离散方法的全球僵硬的隐式式（IMEX）runge-kutta（RK）暂时离散化，并为SCHURECTINET的分散辅助方法，以及SCHURENTECTIONT的整体计算方法，以及SCHUR的整体辅助方法，以及SCHUR的应用。 成本。正式显示了方案的AP特性。通过适用于一阶方案的能量类型稳定性分析，以及应用于第一阶到三阶方案的傅立叶类型稳定性分析，我们确认了该方法相对于$ \ varepsilon $的统一稳定性以及扩散状态中无条件的稳定性。提出了一系列数值示例，以证明新方案的性能。
We consider a linear kinetic transport equation under a diffusive scaling, that converges to a diffusion equation as the Knudsen number $\varepsilon\rightarrow0$. In [3, 21], 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 (RK) temporal discretizations with a new IMEX strategy, discontinuous Galerkin (DG) 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.
