## 3 The Method of Moments  
The conservation laws for the mass, momentum, and energy read [27]:  
\[\left\{ \begin{array}{l}\frac{\partial\rho}{\partial t}+\frac{\partial\rho u _{i}}{\partial x_{j}}=0,\\ \frac{\partial\rho u_{i}}{\partial t}+\frac{\partial\rho u_{i}}{\partial x_{ j}}+\frac{\partial\alpha_{ij}}{\partial x_{j}}=-\frac{\partial\rho}{\partial x_{ i}}+\rho g_{i},\\ \frac{\partial\rho T}{\partial t}+\frac{\partial\rho u_{i}T}{\partial x_{i}}+ \frac{2}{3R}\frac{\partial q_{i}}{\partial x_{i}}=-\frac{2}{3R}\bigg{(}p\frac {\partial u_{i}}{\partial x_{i}}+\sigma_{ij}\frac{\partial u_{j}}{\partial x_ {i}}\bigg{)},\end{array} \right. \tag{8}\]  
in which, the pressure \(p\) is related to the temperature \(T\) by the ideal gas law \(p=\rho RT\). \(\sigma_{ij}\) and \(q_{i}\) are the stress tensor and heat flux, respectively. Any subscript \(i\), \(j\), \(k\) represents the usual summation convention. The classic way to close this set of equations is through a CE expansion of the molecular distribution function, and yields the NSF equations:  
\[\sigma_{ij}=\sigma_{ij}^{NSF}=-2\mu\frac{\partial u_{<i}}{\partial x_{j}},\text { and }q_{i}=q_{i}^{NSF}=-\frac{15}{4}R\mu\frac{\partial T}{\partial x_{i}}, \tag{9}\]  
in which, the angular brackets are used to denote the traceless part of a symmetric tensor. With the moments obtained from the NSF solutions, i.e., \(\left\{\rho,u_{i},T,\sigma_{ij},q_{i}\right\}\), the VDF can be approximated by the first-order expansion of Hermite polynomials \(f^{(1)}\),  
\[f^{(1)}=f_{M}\bigg{[}1+\frac{\sigma_{ij}c_{i}c_{j}}{2 pRT}+\frac{c_{i}q_{i}}{pRT }\bigg{(}\frac{c^{2}}{5 RT}-1\bigg{)}\bigg{]}. \tag{10}\]  
As the value of the Knudsen number increases, [PERSON]'s law for stress and strain and [PERSON]'s law for heat flux and temperature gradient in Equation (9) do not hold anymore, and it is necessary to adopt the method of moments. In R13 moment method, the governing equations for the \(\sigma_{ij}\) and \(q_{i}\) are:  
\[\left\{ \begin{array}{l}\frac{\partial\sigma_{ij}}{\partial t}+\frac{ \partial u_{k}\sigma_{ij}}{\partial x_{k}}+\frac{\partial m_{jk}}{\partial x _{k}}&=-\frac{p}{\mu}\sigma_{ij}-2p\frac{\partial u_{<i}}{\partial x_{j}}- 4\frac{\partial u_{<i}}{5}\frac{\partial u_{<i}}{\partial x_{j}}-2\sigma_{ k<i}\frac{\partial u_{j}}{\partial x_{k}},\\ \frac{\partial q_{i}}{\partial t}+\frac{\partial u_{j}q_{i}}{\partial x_{j}}+ \frac{1}{2}\frac{\partial R_{ij}}{\partial x_{j}}&=-\frac{2}{3}\mu-\frac{2}{ 5}pR\frac{\partial T}{\partial x_{i}}-\frac{7\sigma_{R}g_{R}}{2\alpha_{R}} \frac{\partial T}{\partial x_{k}}-RT\frac{\partial\sigma_{ik}}{\partial x_{k} }+\frac{\sigma_{ij}}{\partial x_{k}}\bigg{)}\\ &-\frac{2}{5}\Big{(}2q_{k}\frac{\partial u_{i}}{\partial x_{k}}+q_{k}\frac{ \partial u_{k}}{\partial x_{i}}+q_{i}\frac{\partial u_{k}}{\partial x_{k}} \Big{)}-\frac{1}{6}\frac{\partial\Delta}{\partial x_{i}}-m_{ijk}\frac{\partial u _{j}}{\partial x_{k}},\end{array} \right. \tag{11}\]respectively. In which \(m_{ijk},\ R_{ij},\ \Delta\) represent the unknown moments. To closed this set of R13 moment equations, the algebraic expressions for \(m_{ijk},\ R_{ij},\ \Delta\) have been derived by [PERSON] and [PERSON] [27; 36]. An alternative way to evaluate these high order moments \(m_{ijk},\ R_{ij},\ \Delta\) is by deriving their governing equations from the Boltzmann equation. In R26 moment method, they are [42]:  
\[\left\{