## 2 The Kinetic Equation and Discrete Velocity Method  
From the microscopic point of view, the behavior of the rarefied gas dynamics can be described from the Boltzmann model equation:  
\[\frac{\partial f}{\partial t}+C_{i}\frac{\partial f}{\partial x_{i}}+g_{i} \frac{\partial f}{\partial C_{i}}=-\frac{1}{\tau}[f-f^{eq}], \tag{1}\]  
where \(f=f(t,x_{i},C_{i})\) denotes the VDF of gas molecular with the molecular velocity \(C_{i}\) at the position \(x_{i}=(x,y,z)\) and the time \(t\). \(g_{i}\) is the external acceleration. The mean relaxation time \(\tau\) is related to the viscosity \(\mu\) and pressure \(p\) by \(\tau=\mu/p\)[50]. In this work, the Bhatnagar-Gross-Krook (BGK) collision model [15] is utilized, and the equilibrium VDF can be defined as [40]:  
\[f^{eq}=f_{M}=\frac{\rho}{\left(2\pi RT\right)^{3/2}}\exp\left(-\frac{\left(C_{ i}-u_{i}\right)^{2}}{2 RT}\right), \tag{2}\]  
where \(\rho\), \(u_{i}\) and \(T\) represent the density, bulk velocity and the temperature of the gas, respectively. \(R\) is the gas constant. For convenience, the intrinsic or peculiar velocity is introduced as, \(c_{i}=C_{i}-u_{i}\). The classical way to solve this Boltzmann model equation is through the time-implicit discrete velocity method. Firstly, the time derivate term can be removed, and the discretization of the partial differential equation can be expressed by:  
\[C_{i}\frac{\partial f^{n+1}}{\partial x_{i}}+\frac{1}{\tau}f^{n+1}=\frac{1}{ \tau}f^{eq,n}-g_{i}\frac{\partial f^{n}}{\partial C_{i}}, \tag{3}\]  
where the superscript \(n\) and \(n+1\) stand for the \(n\)th and \(n+1\)th iterations, respectively. The right-hand side of the Equation (3) is the explicit part, and the left-hand side is the implicit part which can be solved by introducing a second-order upwind scheme. In the upwind scheme, the spatial gradient term with respect to the mesh point \(x=x_{i}\) (e.g., in the x-direction), is evaluated by:  
\[\left\{ \begin{array}{l}\left.\frac{\partial f^{n+1}}{\partial x}\right|_{x=x _{i}}=\frac{3f_{i-1}^{n+1}-4f_{i-1}^{n+1}+f_{i-2}^{n+1}}{2\Delta x},C_{x}>0\\ \left.\frac{\partial f^{n+1}}{\partial x}\right|_{x=x_{i}}=\frac{-f_{i-2}^{n+1 }+4f_{i-1}^{n+1}-3f_{i-1}^{n+1}}{2\Delta x},C_{x}<0\end{array} \right.. \tag{4}\]  
Then, the evolution equations for the VDF can be obtained by substituting Equation (4) into Equation (3), which yields:  
\[\left\{ \begin{array}{l}\left[\frac{1}{\tau}+\frac{3C_{x}}{2\Delta x}\right] f_{i}^{n+1}=\frac{1}{\tau}f_{i}^{eq,n}-g_{x}\frac{\partial f_{i}^{n}}{\partial C _{x}}+C_{x}\frac{4f_{i-1}^{n+1}-f_{i-2}^{n+1}}{2\Delta x},C_{x}>0\\ \left[\frac{1}{\tau}-\frac{3C_{x}}{2\Delta x}\right]f_{i}^{n+1}=\frac{1}{\tau} f_{i}^{eq,n}-g_{x}\frac{\partial f_{i}^{n}}{\partial C_{x}}+C_{x}\frac{f_{i+1}^{n+1}- 4f_{i-1}^{n+1}}{2\Delta x},C_{x}<0\end{array} \right.. \tag{5}\]Meanwhile, the Maxwell's kinetic diffuse-specular wall boundary condition is applied [40], which assumes that a fraction \(\alpha\) of gas molecules undergoes diffusive reflection with Maxwellian distribution function \(f_{M}^{\alpha}\), while the remaining part \((1-\alpha)\) will be reflected specularly. Suppose the wall has a temperature of \(T^{\alpha}\) and moves with the velocity \(u^{\alpha}\), the VDF for the reflected molecules at the wall is given by:  
\[f^{\alpha}= \begin{cases}\alpha f_{M}^{\alpha}+(1-\alpha)f(-C_{i}n_{i}),&C_{i} n_{i}\geq 0,\\ f(C_{i}n_{i}),&C_{i}n_{i}<0,\end{cases}  \tag{6}\]  
where \(n_{i}\) is the normal unit vector of the wall pointing towards the gas in a frame in which the coordinates are attached to the wall. The reflected Maxwellian distribution function can be expressed as:  
\[f_{M}^{\alpha}=\frac{\rho^{\alpha\
u}}{\sqrt[3]{2\pi RT^{\alpha}}}\exp(-\frac {\left|C_{i}-u_{i}^{\alpha}\right|^{2}}{2 RT^{\alpha\
u}}), \tag{7}\]  
where \(\rho^{\alpha\
u}\) is the reflected gas density and need to be determined such that the mass flux across the wall is equal to zero.