## 3 Numerical Results
### Nonlinear Landau Damping  
In linear Landau damping case discussed above, the non-equilibrium phenomenon is still not very significant because the initial perturbation of density is very small. In this example, we increase amplitude of the initial perturbation of density in 1D space. The perturbation rate is set to be \(\alpha=0.5\). The wave number and periodic length remain the same \(k=0.5\) and non dimensional scale of axis \(L=4\pi foraperiodindisturbance\). The initial particle distribution function reads:  
\[f=\frac{1}{\sqrt{2\pi}}e^{-\frac{\pi^{2}}{2}}\left(1+\alpha\cos\left(kx\right) \right). \tag{32}\]  
Figure 3: Evolution of electric potential energy of the linear Landau damping with non dimensional time. Figure 2: Evolution of symmetric electric field of the linear Landau damping with non dimensional time. In UGKS, the discrete particle velocity space is applied. The 1D velocity space is truncated at \([-6,6]\) with 128 intervals. When the velocity space is being discretized, we used cosine to decide the positions of discrete velocity points so that a high resolution can be obtained near the peak of distribution function. This equation is written as  
\[u_{x}=-v_{\max}\left(1+\cos\left(-\frac{\pi}{2}+\pi\frac{i}{Nu-1}\right) \right), \tag{33}\]  
where \(v_{\max}\) is the maximum of particle velocity and \(Nu\) is the number of discrete velocities of particle. The non dimensional scale of microscopic velocity space is decided by \(U/C\), in which \(C\) denotes speed of sound. The evolution of the kinetic entropy is taken as a benchmark solution. Traditional FEMs always overestimate this value.The particle-based methods may be an alternative, but computational cost would be overwhelming. The UGKS is able to obtain the same accuracy as particle-based method but calls for much lower computational cost. The comparison result between UGKS and [PERSON] et al. [12] is presented in Figure 4. The kinetic entropy is computed by \(H=-\sum f\ln f\). In evolution, non dimensional numerical time is decided by \(t/t^{*}\). Reference time \(t^{*}=L_{r}ef/C\), in which \(L_{r}ef=4\pi\). Another benchmark result is \(L^{2}\)-norm of the distribution function. Variation of \(L^{2}\)-  
The results of UGKS fit well with the ones of the particle-based method. With a much lower computational cost, the UGKS can archive the same accuracy. The UGKS has a good conservation property and good accuracy in reconstruction, so that the spurious dissipation is avoided. The nonlinear Landau damping case is a highly non-equilibrium phenomenon. To show this, particle distribution functions at different time steps at the center of space are presented in Figure 6.  
Figure 6 shows the distribution function in velocity space. To give a better description of particle distribution in phase space, the distribution function in the \((x,u)\) space is presented in Figure 7. Distribution functions at \(x=0\) for \(t=10,20,30\) are given in Figure 8 to compare with the results in Reference [35]. The heaps of distribution profiles fit with each other. However, difference between [PERSON]'s equation in this paper and complete [PERSON] equation causes inconsistent results. To simplify computation and reduce computational cost, only electric Poisson's equation is considered. However, in simulation of plasma in electric field, it is an accurate method. In nonlinear Landau damping case, nonlinear effect dominates the evolution of particle motion and electric field. UGKS successfully captures the unsteady and highly nonlinear waves in plasma. Numerical scheme proposed in this paper provides an accurate method for highly non-equilibrium plasma flow simulation. Time development of entropy under meshes with different cell size \(N_{x}\) and discrete velocity space with different number of points \(N_{u}\) are given in Figure 9. The computation results convergence at \(N_{x}=128\) and \(N_{u}=128\) for present method. Figure 5: Evolution of \(L^{2}\)-norm of distribution function in nonlinear Landau damping with non dimensional time.