## 3 Numerical Results
### Gaussian Beam
\begin{array}{cc}\rho\frac{\lambda}{\pi}e^{-\lambda\left((u-U) ^{2}+(v-V)^{2}\right)}&x^{2}+y^{2}\leq r_{a}^{2},\\ 0&x^{2}+y^{2}>r_{a}^{2}.\end{array} \right. \tag{40}\]  
In kinetic theory, \(\lambda\) can be obtained by [56]  
\[\lambda=\frac{m}{2k_{B}T}, \tag{41}\]  
where \(m\) is the particle mass, \(k_{B}\) is the Boltzmann constant, \(T\) is the temperature. The mesh for beam case has a round boundary, which is shown in Figure 10. In this case, 4032 mesh cells are used to discretize the computational domain. Absorbing boundary condition (ABC) is applied in Gaussian beam case. The buffer zone is added between the computational domain and boundary with a range \(0.5^{2}<x^{2}+y^{2}<1.0^{2}\). The flux damps along warp of a circular zone in buffer zone. \[F_{r_{o}}=\delta F_{r_{i}}, \tag{42}\]  
where \(\delta\) is damping rate, \(r_{o}\) and \(r_{i}\) are the two location on a warp line respectively. The damping rate can be computed as  
\[\delta=\cos\left(\frac{\pi}{2}\frac{r-r_{i}}{r_{o}-r_{i}}\right), \tag{43}\]where \(r_{i}\) and \(r_{o}\) represent the two limit of buffer zone. In our work, we have  
\[r_{i} =0.5, \tag{44}\] \[r_{o} =1.0.\]  
The location for \(F_{r_{o}}\) and \(F_{r_{i}}\) is described in Figure 11.  
In evolution, the electric field induced by charged particles and particle free transport will expand the beam. Applied external electric field focalizes the beam. So there is a process in which expansion and contraction take place alternately. We describe evolution process of beam using electron density contour at different time steps. The particle velocity is truncated at 6.0 with 64 points. We simulated the evolution of a Gaussian beam and compared our results with the work of Reference [12], which proves the accuracy of UGKS in plasma flow simulation. Figure 11: Buffer zone and flux on the warp for simulation of the Gaussian beam under non dimensional physical space. Figure 10: Mesh for the simulation of the Gaussian beam under non dimensional physical space. Now, let's compare the computational cost with particle-based methods introduced in Reference [12]. The test model is put in a \(64\times 64\times 64\times 64\) phase space. We compared computation time of the UGKS and particle-based scheme under the same phase space in Table 1. The comparison results show a high efficiency of UGKS under the same condition. Although electron has wave-particle dualism, in this paper, we only study Gaussian beam under an electrodynamics frame. Based on the Boltzmann-Poisson system, the motion of electrons is described in phase space with particle distribution function. Electrons are moving in space for existence of electromagnetic force. So the peak value of intensity of the beam varies with time in Figure 10.0 In evolution, non dimensional numerical time is decided by \(t/t^{*}\). Reference time \(t^{*}=L_{r}ef/C\), in which \(L_{r}ef=2r_{a}\). To obtain a correct evolution process of beam, the transport and acceleration term in the Boltzmann equation should be accurately coupled. Although the wave nature for electrons is out of the range in this paper, the Schrodinger equation can be used to construct a numerical method for quantum results of Gaussian beam case in the future. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Processors** & **Unified Gas Kinetic Scheme** & **PIC** \\ \hline
1 & 2645.7 s & 11,324.1 s \\
2 & 1458.6 s & 6287.0 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computation time for comparison under \(64\times 64\times 64\times 64\) phase space. PIC: particle-in-cell. Figure 12: _Cont._