The flow regime of micro flow varies from collisionless regime to hydrodynamic regime according to the Knudsen number Kn, which is defined as the ratio of the mean free path over the local characteristic length. On the kinetic scale, the dynamics of a small-perturbed micro flow can be described by the linearized kinetic equation. In the continuum regime, according to the Chapman-Enskog theory, hydrodynamic equations such as linearized Euler equations and Navier-Stokes equations can be derived from the linearized kinetic equation. In this paper, we are going to propose a unified gas kinetic scheme (UGKS) based on the linearized kinetic equation. For the simulation of small-perturbed micro flow, the linearized scheme is more efficient than the nonlinear one. In the continuum regime, the cell size and time step of UGKS are not restricted to be less than the particle mean free path and collision time, and the UGKS becomes much more efficient than the traditional upwind-flux-based operator-splitting kinetic solvers. The important methodology of UGKS is the following. Firstly, the evolution of microscopic distribution function is coupled with the evolution of macroscopic flow quantities. Secondly, the numerical flux of UGKS is constructed based on the integral solution of kinetic equation, which provides a genuinely multiscale and multidimensional numerical flux. The UGKS recovers the solution of linear kinetic equation in the rarefied regime, and converges to the solution of the linear hydrodynamic equations in the continuum regime. An outstanding feature of UGKS is its capability of capturing the accurate viscous solution in bulk flow region once the hydrodynamic flow structure can be resolved by the cell size even when the cell size is much larger than the kinetic length scale, such as the capturing of the viscous boundary layer with a cell size being much larger than the particle mean free path. Such a multiscale property is called unified preserving (UP) which has been studied in (Guo, et al. arXiv preprint arXiv:1909.04923, 2019). In this paper, a mathematical proof of the UP property for UGKS will be presented and this property is applicable to UGKS for solving both linear and nonlinear kinetic equations.

中文翻译：

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基于线性动力学方程的微流统一气体动力学方案

微流的流态根据Knudsen数Kn从无碰撞态到流体力学态变化，后者定义为平均自由程与局部特征长度之比。在动力学尺度上，可以通过线性化动力学方程来描述小扰动微流的动力学。在连续状态下，根据Chapman-Enskog理论，可以从线性动力学方程派生流体动力学方程，例如线性Euler方程和Navier-Stokes方程。在本文中，我们将基于线性动力学方程式提出统一的气体动力学方案（UGKS）。对于微扰动微流的仿真，线性化方案比非线性方案更有效。在连续统体制下，不限制UGKS的像元大小和时间步长小于粒子平均自由程和碰撞时间，并且UGKS的效率要比传统的基于迎风通量的算符分裂动力学求解器高得多。UGKS的重要方法如下。首先，微观分布函数的演化与宏观流量的演化耦合。其次，基于动力学方程的积分解构造了UGKS的数值通量，提供了真正的多尺度和多维数值通量。UGKS恢复稀疏状态下线性动力学方程的解，并收敛到连续状态下线性流体动力学方程的解。UGKS的一个突出特点是，一旦孔的大小能够解决流体动力流动结构的问题，即使孔的尺寸远大于动力学长度尺度，它也可以在大流量区域中捕获准确的粘性溶液，例如单元格大小远大于粒子平均自由程的粘性边界层。这种多尺度属性称为统一保存（UP），已在（Guo等人，arXiv预印本arXiv：1909.04923，2019）中进行了研究。本文将给出UGKS UP性质的数学证明，该性质适用于UGKS求解线性和非线性动力学方程。例如捕获比细胞平均自由程大得多的孔尺寸的粘性边界层。这种多尺度属性称为统一保存（UP），已在（Guo等人，arXiv预印本arXiv：1909.04923，2019）中进行了研究。本文将给出UGKS UP性质的数学证明，该性质适用于UGKS求解线性和非线性动力学方程。例如捕获比细胞平均自由程大得多的单元尺寸的粘性边界层。这种多尺度属性称为统一保存（UP），已在（Guo等人，arXiv预印本arXiv：1909.04923，2019）中进行了研究。本文将给出UGKS UP性质的数学证明，该性质适用于UGKS求解线性和非线性动力学方程。