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The flow regime of micro flow varies from collisionless regime tohydrodynamic regime according to the Knudsen number. On the kinetic scale, thedynamics of micro flow can be described by the linearized kinetic equation. Inthe continuum regime, hydrodynamic equations such as linearized Navier-Stokesequations and Euler equations can be derived from the linearized kineticequation by the Chapman-Enskog asymptotic analysis. In this paper, based on thelinearized kinetic equation we are going to propose a unified gas kineticscheme scheme (UGKS) for micro flow simulation, which is an effectivemultiscale scheme in the whole micro flow regime. The important methodology ofUGKS is the following. Firstly, the evolution of microscopic distributionfunction is coupled with the evolution of macroscopic flow quantities.Secondly, the numerical flux of UGKS is constructed based on the integralsolution of kinetic equation, which provides a genuinely multiscale andmultidimensional numerical flux. The UGKS recovers the linear kinetic solutionin the rarefied regime, and converges to the linear hydrodynamic solution inthe continuum regime. An outstanding feature of UGKS is its capability ofcapturing the accurate viscous solution even when the cell size is much largerthan the kinetic kinetic length scale, such as the capturing of the viscousboundary layer with a cell size ten times larger than the particle mean freepath. Such a multiscale property is called unified preserving (UP) which hasbeen studied in \cite{guo2019unified}. In this paper, we are also going to givea mathematical proof for the UP property of UGKS.

A Unified Gas-kinetic Scheme for Micro Flow Simulation Based on Linearized Kinetic Equation