## 5 Numerical Results and Discussions
### Lid-Driven Cavity Flow
In terms of the \(\overline{u}_{y}\), the R26 and R13 moment equations overpredict and underpredict the velocity adjacent to the wall by about 4.8% and 5.4%, respectively. It is also interesting to find that the heat flux profiles predicted by the R26 moment equations are not as accurate as these predicted by the R13 moment equations at \(Kn=0.1\). Similar phenomena appear in the pressure-driven Poiseuille flow given by [PERSON] and [PERSON] (2019), which demonstrates that the R26 moment equations are not necessary more accurate than the R13 moment equations. As the Knudsen number further increase, e.g., \(Kn=0.5\), all of the macroscopic equations fail to reproduce correct heat flux profiles. That is because, compared to the lower-order moment quantity, such as the velocity, the heat flux is more sensitive to the accuracy of the VDF. However, both R13 and R26 moment equations are able to predict relatively accurate velocity profiles along the centerlines of the cavity, as shown in Figure 8. When the Knudsen number is above 1, it will be very difficult for the R13 and R26 moment equations to find the steady-state solutions. Figure 7: The simulation results of the cavity flow at \(Kn=0.1\). In this case, three Knudsen numbers, i.e., \(Kn=0.01\), \(Kn=0.1\), \(Kn=0.5\), are considered. The errors of the reconstructed VDFs are compared and presented in Figure 9.  
Several interesting phenomena can be observed from Figure 9. Firstly, as the Knudsen number increases (from \(Kn=0.01\) to \(Kn=0.5\), as shown in each column of the figure), all of the errors \(E_{s}^{(1)},E_{s}^{(3)},E_{s}^{(5)}\) increase. Secondly, from each row of the figure, we can see the overall solutions indicate that \(E_{s}^{(1)}>E_{s}^{(3)}>E_{s}^{(5)}\), which demonstrates that the R26 moment equations are more accurate than the R13 moment equations and the NSF equations at the microscopic level. Last but not least, from each sub-figure, we can see that the maximum errors appear near the upper wall of the cavity, especially at the corners of the cavity. In contrast, the minimum errors appear near the bottom of the cavity. Take \(Kn=0.5\) as an example, we have compared \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) near the upper and bottom walls to evaluate whether the 'nonequilibrium effects' near the upper wall are stronger than that near the bottom wall. Two characteristic points include \((\overline{x}=0,\ \overline{y}=-0.48)\) and \((\overline{x}=0,\ \overline{y}=0.48)\) are chosen along the vertical centerline of the cavity. The comparison of the \(\overline{f}\) with \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) are shown in Figure 10, and they are evaluated under the same scale in \(\overline{C}_{x}\)-direction. It can be seen that the difference between the \(\overline{f}\) and \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) in Figure 10b is larger than that in Figure 10a, which means the 'nonequilibrium effects' in the upper wall region is stronger than that in the bottom wall region. Besides, we also find that both \(\overline{f}^{(3)}\) and \(\overline{f}^{(5)}\) are still very close to the Maxwellian equilibrium VDF, and that is why the R13 and R26 moment methods are not working when the Knudsen number is above 0.5. For this case, all tests are done on a single processor, the computational memory and time cost are listed in Figure 6.0 The convergence criterion for the steady-state is defined by Equation (19). We can see that the R13 and R26 moment methods have the ability to save the computational memory and time cost, especially in the slip and early transition regimes. Figure 6.1 The simulation results of the cavity flow at \(Kn=0.5\). Figure 6.2 Comparison of the VDF \(\overline{f}\) with \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\).