## 1 Introduction
(2015) simulated the Navier-Stokes dielectric barrier discharge (NS-DBD) plasma formation between parallel electrodes in N\({}_{2}\)-O\({}_{2}\) mixture air at low-pressure under nanosecond impulses [15]. However, in these schemes, individual particle motion is resolved in flux evaluation so numerical time step is limited within relaxation time. In current paper, a unified gas kinetic scheme (UGKS) is applied to simulate plasma flows based on the Boltzmann-Poisson equation. The UGKS is proposed by [PERSON], [PERSON] and [PERSON] [16; 17], which attracts more and more researchers' attention [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. In plasma flow, particle transport is controlled by electric force, so the state of gas is decided by electric field. In mathematics, electric force term does not appear in advection part but a source term in discretization. Gas distribution function can never evolve towards Maxwellian as the Bhatnagar-Gross-Krook (BGK) model describes. The real gas distribution function in the highly non-equilibrium region is never able to be described by a Maxwellian. In this paper, the particle velocity space is discretized so that distribution can be exactly described in all flow regimes [16]. The flux reconstruction at the cell interface is treated as the Riemann problem along characteristic curves presented as particle transport velocity [33]. The free transport and interaction between particles are coupled so that the dissipation in the transport process is controlled by the source of the long-range interaction rather than numerical time step. With this technique, the artificial dissipation as the one in [PERSON] et al. (2011) [14] will not be brought in numerical process. With integral solution of Boltzmann equation, The UGKS obtains accurate gas distribution function at cell interface at each time step [33]. The numerical time step can be decided by the Courant- Friedrichs-Lewy (CFL) condition [34]. This high fidelity and efficiency in numerical simulation make the UGKS a better way to solve plasma problems. Based on the BGK model equation, [PERSON] et al. (2017) simulated multi-scale and multi-component plasma transport based on the UGKS [35]. Many challenging cases including magnetic reconnection problem in the transition regime are simulated. In study of plasma, UGKS provides a reliable multi-scale approach for numerical simulation. In solving the Boltzmann equation, gas distribution function at cell interface is obtained by the solution for the local Riemann problem [10] by using difference biased in the direction determined by the sign of the characteristic speeds [36], thus UGKS has good conservation property and good robustness. Spectral methods has a high order accuracy in smooth regime [37]. However, highly non-equilibrium state will be a challenge because a large error appears in a strong discontinuity at cell interface of phase space. This problem becomes severe when micro-electromechanical systems should also be paid enough attention in rarefied regime [38]. In plasma, time-varying electric field causes collective behavior with many degrees of variation [39; 40]. Since the flux of gas distribution function for particles in discrete microscopic velocity space is computed in UGKS, the multi-scale property is satisfied. Therefore, the present UGKS is a promising scheme for plasma simulation. Unlike neutral gas, plasma consists of a significant number of charge carriers, which makes it electrically conductive. Because of this feature, plasma responds strongly to electromagnetic field. Plasma does not have certain shape or certain volume if not enclosed in a container, but its motion and state are able to be controlled by electromagnetic force. Under influence of electromagnetic field, plasma can be formed into various structures. An important property used to describe the electric field is electric potential. It represents the amount of electrical energy of a unitary point charge at any point in an electric field.