## 5 Numerical Results and Discussions  
### Force-Driven Poiseuille Flow  
The performance of the moment equations and the DVM is first evaluated by simulating the one-dimensional (1D) force-driven Poiseuille flow between two parallel plates, which are located at \(y=-L_{0}/2\) and \(y=L_{0}/2\). The reference characteristic length and the reference temperature are set to be \(L_{0}=10^{-5}\)m and \(T_{0}=273\) K, respectively. An external force is applied in the \(x\)-direction, and the non-dimensional force is set to be \(\overline{g}_{i}=L_{0}g_{i}/(2 RT_{0})=0.01\). The spatial space is discretized with 51 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 is already validated. In the DVM, the diffuse boundary condition is applied on both plates. Suppose the magnitude of the external acceleration is very small, the derivation of the force term can be approximated by:  
\[g_{i}\frac{\partial f}{\partial C_{i}}\approx g_{i}\frac{\partial^{eq}}{ \partial C_{i}} \tag{18}\]  
The normalized velocity profiles along the channel cross-section using the NSF, R13, R26 equations and the DVM are shown in Figure 1. A good agreement can be observed between the DVM and the macroscopic equations when \(Kn=0.01\), the dimensionless maximum velocity \(\overline{u}_{x}=0.229\) appears in the center of the channel and both the micro- and macroscopic approaches have the ability to capture the velocity profiles accurately. However, when \(Kn=0.1\), the NSF equations in association with the velocity slip and temperature jump boundary conditions underestimate the maximum velocity by about 8.2% in the center of the channel. Meanwhile, the R26 moment equations are more accurate than the R13 equations, which reduces the maximum velocity error from 3.6% to 1.1%. As the Knudsen number further increases, e.g., \(Kn=0.5\) and \(Kn=1\), the NSF equations fail to predict the velocity profiles. Both R13 and R26 moment equations begin to lose their accuracy in the velocity prediction at \(Kn=0.5\), and they underestimate the maximum velocity by about 6.4% and 5.2% in the center of the channel and overpredict the slip velocity by about 28.8% and 12.4% in the near-wall region. From the Figure 1c,d, it is clear to see that the R26 moment equations can reproduce relatively more accurate slip velocity than that predicted by the R13 equations. A detailed comparison for \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) are shown in Figure 2. We choose the point \((\overline{y}=0.48)\) as the characteristic point to evaluate their accuracy. Since the \(f(t,x_{i},C_{i})\) is a multi-dimensional function, the contours in the left panel of Figure 2 are non-dimensional VDFs \(\overline{f}=f(2 RT_{0})^{3/2}/\rho_{0}\) with the molecular velocity \(C_{z}=0\). Similarly, the profiles in the right panel are dimensionless VDFs with the molecular velocity \(C_{y}=0,\ C_{z}=0\). It is clear to see that, as the Knudsen number increases, the VDF shifts away from the equilibrium state. In the slip regime, where \(Kn<0.1\), the VDFs \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) obtained from NSF equations and R13, R26 moment equations agree very well with that obtained from the DVM. However, when \(Kn=1\), the non-equilibrium effects begin to dominate, and \(\overline{f}\) is clearly away from the equilibrium, in contrast, all of the VDFs \(\overline{f}^{(1)},\overline{f}^{(3)},\overline{f}^{(5)}\) are not accurate anymore, as indicated in Figure 2f. From the magnification of the Figure 2f, we can also see that \(\overline{f}^{(5)}\) is closer to the DVM result in comparison with \(\overline{f}^{(3)}\) and \(\overline{f}^{(1)}\). That also helps to explain why the R26 moment method is more accurate than the NSF and the R13 moment method.