The kinetic theory provides a physical basis for developing multiscal methods for gas flows covering a wide range of flow regimes. A particular challenge for kinetic schemes is whether they can capture the correct hydrodynamic behaviors of the system in the continuum regime (i.e., as the Knudsen number $\epsilon\ll 1$ ) without enforcing kinetic scale resolution. At the current stage, {the main approach to analyze such a property is the asymptotic preserving (AP) concept, which aims to show whether a kinetic scheme reduces to a solver for the hydrodynamic equations as $\epsilon \to 0$, such as the shock capturing scheme for the Euler equations. However, the detailed asymptotic properties of the kinetic scheme are indistinguishable when $\epsilon$ is small but finite under the AP framework}. In order to distinguish different characteristics of kinetic schemes, in this paper we introduce the concept of unified preserving (UP) aiming at assessing asmyptotic orders of a kinetic scheme by employing the modified equation approach and Chapman-Enskon analysis. It is shown that the UP properties of a kinetic scheme generally depend on the spatial/temporal accuracy and closely on the inter-connections among the three scales (kinetic scale, numerical scale, and hydrodynamic scale) and their corresponding coupled dynamics. Specifically, the numerical resolution and specific discretization of particle transport and collision determine the flow physics of the scheme in different regimes, especially in the near continuum limit. As two examples, the UP methodology is applied to analyze the discrete unified gas-kinetic scheme and a second-order implicit-explicit Runge-Kutta scheme in their asymptotic behaviors in the continuum limit.
翻译:动能理论为开发涵盖多种流动制度的气体流动多角度方法提供了一个物理基础。动能机制的一个特殊挑战是,它们能否在不强制实施动能比例尺分辨率的情形下(即Knudsen nudsen nu= $\epsilon\ll 1美元)捕捉系统正确的流体动力行为(即Knudsen nudsen nu= $\ epsilon\ll 1美元),而没有执行动能比例尺分辨率。在现阶段,{分析这种属性的主要方法是无运动保存(AP)概念,目的是显示一个动能机制是否降低为流体动力方程式的解析剂,如$\epsilon至0美元,如Eulner等方程式的震荡捕捉系统。然而,当 $\\epslon number $虽小,但在AP 框架下, 却有限 。 {主要的方法是:为了区分动能机制的不同特征,在本文件中,我们提出一个统一保存(UP)概念,目的是通过使用经修改的对流体-直流体物理学流和直径-Enal-Enal-Enalalalal-alalalalal roal roal rout rout rout rol laction lax lax laction lax lax lax lax lax lax lautt lax lax lax lax lax lax lax lax lax lax