Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on Schur complement
Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on Schur complement
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DOI：
10.1137/20m134486x
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发表时间：
2020-06
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影响因子：
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通讯作者：
Zhichao Peng;Fengyan Li
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文献类型：
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作者：
Zhichao Peng;Fengyan Li
我们考虑一个线性动力学输运方程下的扩散标度，收敛到扩散方程的克努森数$\varepsilon\rightarrow0$。在文献[3，21]中，为了在$\varepiral\ll 1$的扩散区域中实现渐近保持（AP）性质和无条件稳定性，基于方程的偶奇或微观-宏观分解版本的附加重新表述开发了数值格式。其关键是在分解后的系统中，在一个方程的两侧增加一个加权扩散项。权函数的选择，但是，是问题相关的和特设的，它可以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，并仍然获得AP属性和无条件稳定的扩散制度，本文提出了一个新的家庭的AP计划，称为IMEX-DG-S计划，直接解决微观-宏观分解系统没有任何进一步的改造。的IMEX-DG-S计划的主要成分包括全球刚性精确隐式显式（IMEX）龙格库塔（RK）时间离散与一个新的IMEX策略，不连续的Galerkin（DG）空间离散，离散纵坐标方法的速度空间，和Schur补的代数形式的计划，以控制总的计算成本的应用程序。形式化地证明了方案的AP性质。通过对一阶格式进行能量型稳定性分析，对一至三阶格式进行傅里叶型稳定性分析，我们确认了方法关于$\varepsilon$的一致稳定性和扩散区域的无条件稳定性。一系列的数值例子来证明新方案的性能。
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.