## 5 Numerical Results and Discussions
### D Thermal Cavity Flow
From the microscopic point of view, like the other cases in this study, \(E_{s}^{(1)},E_{s}^{(3)},E_{s}^{(5)}\) are evaluated at different Knudsen numbers, shown in Figure 14. Several interesting phenomena can be found: Firstly, the overall results show that, \(E_{s}^{(1)}>E_{s}^{(3)}>E_{s}^{(5)}\), which demonstrate the \(\overline{f}^{(5)}\), reconstructed from the R26 moment equations, is more accurate than \(\overline{f}^{(1)}\) and \(\overline{f}^{(3)}\). Secondly, like the lid-driven cavity case, as the Knudsen number increases, all of the VDFs \(\overline{f}^{(1)}\), \(\overline{f}^{(3)}\) and \(\overline{f}^{(5)}\) begin to lose their accuracy, especially in the near wall region. Last but not least, the maximum errors of \(E_{s}^{(1)},E_{s}^{(3)},E_{s}^{(5)}\) appear near the center of the upper and lower walls, which are also the hottest regions of the walls, as indicated in Figures 11 and 14. Hence, we choose the point \((\overline{x}=0,\ \overline{y}=-0.48)\) close to the hottest center of the lower wall as a characteristic point to evaluate the accuracy of VDFs. The comparison of \(\overline{f}\), \(\overline{f}^{(3)}\), and \(\overline{f}^{(5)}\) are shown in Figure 14. The definitions of the dimensionless \(\overline{f}\) and the descriptions of the contours and the lines are similar to that described in the force-driven Poiseuille case. When \(Kn=0.1\), \(\overline{f}\) moves away from the equilibrium states. However, \(\overline{f}^{(1)},\overline{f}^{(3)}\), and \(\overline{f}^{(5)}\) are still very close to the equilibrium states, as indicated in Figure 15c. From the Figure 15d, we can find that the \(\overline{f}^{(5)}\) is more accurate than that obtained from the R13 moment equations and the NSF equations. When \(Kn=0.5\), \(\overline{f}\) appears the special double peaks phenomenon, while all of the VDFs \(\overline{f}^{(1)},\overline{f}^{(3)}\), and \(\overline{f}^{(5)}\) cannot capture this special phenomenon. Figure 13: Temperature profiles \(\overline{T}=T/T_{0}\) along the vertical center (**a**–**c**) and the horizontal center lines (**d**–**f**) at different Knudsen numbers. Figure 14: Contours of the \(E_{s}^{(1)},E_{s}^{(3)},E_{s}^{(5)}\) at (**a–c**) \(Kn=0.01\); (**d–f**) \(Kn=0.1\); (**g–i**) \(Kn=0.5\). Like the other cases in this study, the computational memory and time cost have been evaluated for the thermally gradients induced case, as shown in Table 3. For this case, all tests are done on single processor (Intel Corporation, Santa Clara, CA, USA, i7-8550u), and the convergence criterion for the thermal gradients induced case is \(\overline{\hat{f}}^{(1)}\). The results are shown in Figure 15.  
Figure 15: Comparison of the VDFs at (**a,b**): \(Kn=0.01\); (**c,d**): \(Kn=0.1\); (**e,f**): \(Kn=0.5\). Left panel of the figure, contours and white lines: \(\overline{\hat{f}}\); black lines: \(\overline{\hat{f}}^{(5)}\); red dash dot lines: \(\overline{\hat{f}}^{(3)}\). The remaining notations have been defined in Figure 11:0  
the steady-state is defined by Equation (20). The macroscopic equations are able to save tremendous computational time and memory cost in comparison with the DVM. Figure 11:1