## 3 Numerical Results  
### Linear Landau Damping  
The Landau damping is the effect of damping of longitudinal space charge waves in plasma [49]. It prevents an instability from developing and creates a region of stability in the parameter space [50]. Energy exchange between electromagnetic wave in phase space and charged particles in plasma is the cause of Landau damping. In evolution, particles interact strongly with the wave [51]. In current work, we simulate the Landau damping based on the UGKS to study the way in which electrons interact and exchange energy with electromagnetic field. This case is used to test the accuracy of the UGKS for solving the Boltzmann equation in describing time-variant electromagnetism parameters such as electric field and electric potential energy. First, an electromagnetic wave should be added to flow field. This wave is caused by an initial disturbance of distribution function [52]. In current work, 2D initial condition is set to  
\[f_{0}=\rho\frac{\lambda}{\pi}e^{-\lambda\left(\left(u-U\right)^{2}+\left(v-V \right)^{2}\right)}\left(1+\alpha\cos\left(k_{x}x\right)\cos\left(k_{y}y\right) \right), \tag{30}\]  
where \(\lambda=0.5\), \(U=V=0\), \(\alpha=0.05\) and the wave numbers \(k_{x}=k_{y}=0.5\) in current problem. In this case, the scales of non dimensional axis of X and Y are set to be \(4\pi\) for a period in disturbance. In the linear Landau damping case, a periodic boundary condition is applied. The four dimensional phase space contains 64 points per dimension. The particle velocity is truncated at 6.0. The periodic boundary condition is chosen for 2D linear Landau damping. The length for computational domain in each dimension is set to be \(4\pi\). The constant time step \(\Delta t\) is chosen as 0.01. The evolution of density contour is presented in Figure 1. The energy of electric field is being carried away by electrons moving at the phase space. To show this process, the evolution of symmetric electric field and electric energy will be given. First, we computed the average of symmetric electric field. The logarithmic scale of it is used for damping process of electric field, which is presented in Figure 2. In evolution, non dimensional numerical time is decided by \(t/t^{*}\). Reference time \(t^{*}=L_{r}ef/C\), in which \(L_{r}ef=4\pi\) and \(C\) denotes speed of sound. Now, the electric potential energy is computed to show the process of energy transport and conversion during interaction between the electric field and electrons. The electric potential energy is a potential energy that results from coulomb forces between charges. This kind of energy is associated with the configuration of a defined system which contains a certain number of charged particles. In physics, the electric potential energy of a system is the energy required for assembling charges from an infinite distance by bringing them close together. If an object has electric potential energy \(U_{E}\), two key elements are crucial: (1) this object keeps its own charge; (2) it stays at a relative position to other electrically charged objects. \[U_{E}=q\Phi. \tag{31}\]  
The evolution of potential energy in systems with time-variant electric fields is given in Figure 3.  
In the current work, we obtain the same evolution process of the electric field and electric energy as Reference [12]. Many works have been done to study the physical picture of Landau damping. Linearized theory is the simplest and rather complete way [53], but it is not physically reasonable because interaction between charged particles and electromagnetic field is coupled with particle transport in the Boltzmann equation. The linearized method is not appropriate to be used in this nonlinear system. However, nonlinearity has been a longstanding problem. To model nonlinear level in the Vlasov equation, a class of exponentially damped solutions of the Vlasov-Poisson equation is proposed [54]. This method fails to explain the mechanism of energy transport although it approximates the damping curves in a mathematical way.