## 6 Conclusions  
The main objective of this work is to quantify the computational performance of the moment methods at both the macro- and microscopic levels, so that researchers may choose the most appropriate method for their applications. Three types of rarefied gas flows have been investigated with different Knudsen numbers and the results have been validated using DVM results. From the macroscopic point of view, the simulation results show that, as the Knudsen number increases, the NSF equations will gradually lose this validation and accuracy. In contrast, both R13 and R26 moment equations are able to reproduce accurate results in the slip and the transition regimes.  
From the microscopic point of view, the errors of the reconstructed VDFs are evaluated based on the 'non-equilibrium indicator'. Several interesting phenomena have been found. Firstly, as the value of the Knudsen number increases, \(E_{s}^{(1)}\), \(E_{s}^{(3)}\) and \(E_{s}^{(5)}\) increase subsequently, especially in the near wall region. Secondly, the overall results show that, \(E_{s}^{(1)}>E_{s}^{(3)}>E_{s}^{(5)}\), which demonstrates the \(\overline{f}^{(5)}\) is more accurate than \(\overline{f}^{(1)}\) and \(\overline{f}^{(3)}\). Last but not least, even though \(\overline{f}^{(5)}\) is much closer to the \(\overline{f}\), all of \(\overline{f}^{(1)},\overline{f}^{(3)}\), and \(\overline{f}^{(5)}\), are still very close to the equilibrium state. In terms of the computational cost, both the R13 and R26 moment methods have the ability to save tremendous computational memory (by more than 95%) and time cost in the slip and early transition regimes, in comparison with the DVM.  
In summary, the moment method bridges the gap between the conventional hydrodynamic model and kinetic model in early transition regime, where the NSF and DVM become either inaccuracy or inefficient. In the slip and early transition (where the Knudsen number is below 0.5), the moment methods can predict the flow field accurately, besides, they are much faster than the implicit DVM, especially when \(Kn\leq 0.1\). However, when \(Kn>0.5\), the time-implicit DVM is a better choice.  
[PERSON] developed the code and performed the simulations. [PERSON] analyzed the results and wrote the paper, [PERSON], [PERSON] helped to revise the paper and analyze the results.  
This research was funded by the National Natural Science Foundation of China (61702538), Scientific Research Projects of the National University of Defense Technology (ZK17-03-53).  
The financial support to [PERSON] from [PERSON] and the Chinese Scholarship Council (CSC) during his visit to the UK are greatly acknowledged. [PERSON] would also like to thank [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] (at the University of Strathclyde, Glasgow, UK) and [PERSON], [PERSON] (STFC Daresbury Laboratory, UK) for the useful discussions in DVM and moment method.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{
\begin{tabular}{c} **Computational** \\ **Memory (MB)** \\ \end{tabular} } & \multicolumn{3}{c}{**Computational Time (Minutes)**} \\ \cline{3-5}  & & _Kn = 0.01_ & _Kn = 0.1_ & _Kn = 0.5_ \\ \hline DVM & 10691 & 265.14 & 126.24 & 36.6 \\ NSF & 10.8 & 0.46 & 1.00 & 1.20 \\ R13 & 11.2 & 1.08 & 5.58 & 15.50 \\ R26 & 11.2 & 1.20 & 10.26 & 24.46 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the computational cost between the DVM and moment method.