## 5 Numerical Results and Discussions
### Lid-Driven Cavity Flow  
In addition to the force-driven Poiseuille flow, the evaluation of the moment method is also performed on a 2D lid-driven cavity flow, which is also a standard benchmark problem to validate numerical accuracy and efficiency. The length and the height of the cavity are both set to be \(10^{-5}\) m, and the velocity of the top-wall \(u_{0}\) is 10 m/s. The reference length, temperature and viscosity are \(L_{0}=10^{-5}\) M, \(T_{0}=273\) K and \(\mu_{0}=21.25\times 10^{-6}\text{Pa}\cdot\text{s}\), respectively. Three Knudsen numbers, i.e., \(Kn=0.01,\ 0.1,\ \text{and}\ 0.5\) are considered. The spatial space is discretized with \(101\times 101\) non-uniform grids, and the velocity space for the DVM is discretized in a range of \(\left[-6\sqrt{2 RT_{0}},\ 6\sqrt{2 RT_{0}}\right]^{3}\) with \(64\times 64\times 24\) non-uniform distributed Newton-Cotes quadrature points. Independence of results with respect to the number of discrete velocities and spatial grids are already validated. In the DVM, the diffuse boundary condition is applied on all the solid walls. Figure 5 shows the normalized velocity profiles \(\overline{u}_{i}=u_{i}/\sqrt{2 RT_{0}}\) and the normalized heat flux \(\overline{q}_{i}=q_{i}/\left(\rho_{0}\left(\sqrt{2 RT_{0}}\right)^{3}\right)\) along centerlines of the cavity for \(Kn=0.01\) in the early slip regime. It is usually recognized that it is difficult for the traditional DVM in this regime due to the requirement of large meshes. However, in terms of the velocity prediction, we note that the results obtained from the microscopic and macroscopic equations with the fine mesh of \(101\times 101\) are in excellent agreement with each other. The dimensionless heat flux \(\overline{q}_{i}\) predicted by both the microscopic and macroscopic methods are presented in Figure 5c,d. It can be seen that the NSF equations with the velocity slip and temperature jump boundary conditions fail to reproduce the heat flux profiles at the Knudsen number of 0.01. Besides, the R13 moment equations overpredict the absolute value of the heat flux adjacent to the wall compared to the R26 moment equations. The overall agreement between the microscopic and macroscopic approaches in terms of the velocity profiles is better than that of the heat flux profiles. That is because, the heat flux, a high-order moment quantity, is more sensitive to the core model than low-order ones. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{
\begin{tabular}{c} **Computational** \\ **Memory (MB)** \\ \end{tabular} } & \multicolumn{4}{c}{**Computational Time (Min)**} \\ \cline{3-6}  & & \(Kn=0.01\) & \(Kn=0.1\) & \(Kn=0.5\) & \(Kn=1\) \\ \hline DVM & 5562.00 & 142.2 & 19.18 & 3.65 & 2.40 \\ NSF & 2.20 & 0.50 & 0.20 & 0.25 & 1.27 \\ R13 & 2.20 & 2.00 & 1.20 & 3.52 & 6.46 \\ R26 & 2.20 & 3.20 & 2.50 & 4.36 & 10.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the computational cost between the DVM and the moment methods. The accuracy of \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) at \((\overline{x}=0,\ \overline{y}=-0.48)\) is shown in Figure 6. It is clear to see that when \(Kn=0.01\), \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) agree very well with the VDF obtained from the DVM. Figure 5: The simulation results of the cavity flow at \(Kn=0.01\). Figure 6: Comparison of the dimensionless VDFs \(\overline{f},\overline{f}^{(1)},\overline{f}^{(3)}\), and \(\overline{f}^{(5)}\) at \(Kn=0.01\). The notations have been defined in Figure 2.  
In the early transition regime, e.g., \(Kn=0.1\), the NSF equations underpredict the velocity by about 12.3% close to the wall, as indicated in Figure 7. The R13 and R26 moment equations represent almost the same \(\overline{u}_{x}\) velocity profiles along the vertical centerline, and both of them agree very well with the DVM results.