## 1 Introduction  
The Boltzmann equation describes time evolution of physical state under external field using gas-distribution function [1]. In description of plasma with long-range Coulomb interaction, [PERSON] showed the difficulties when kinetic theory based on standard transport-collision is applied: (1) Theory of pair collisions disagrees with the discovery by [PERSON], [PERSON] and [PERSON] that vibrations exist in electron plasma; (2) Theory of pair collisions without Coulomb interaction will lead to divergence of kinetic term; (3) Theory of pair collisions cannot explain results of experiments by [PERSON] and [PERSON] that electrons scatter anomalously in gaseous plasma [2]. In gas, binary interaction is taken as the rule. However, in plasma, waves, or organized motion of plasma, are very important because the particles can interact at long ranges through the electric and magnetic forces. [PERSON] suggested a source term which contains the Coulomb interaction added in Boltzmann equation. It is a partial differential equation (PDE) for distribution function of particles, which represents probability that particle stays at a specific position and velocity. The acceleration term in the equation describes redistribution of particle on particle phase space due to electromagnetic force. Many work about numerical methods have been done to model plasma evolution. Particle based method is a direct simulation method by tracing particle trajectory. [PERSON] et al. (2010) proposed a particle-in-cell (PIC) method based on Vlasov equation [3]. In PIC method, the particle details, such as the location, velocity, are exactly captured by the equation of motion. To reduce computational cost, finite element method (FEM) and spectral method are proposed. In FEM the whole computational domain is divided into several discrete elements and a polynomial is applied to approximate the distribution function in phase space. [PERSON] et al. (2011) proposed a Galerkin method for the Vlasov equation. Lagrange polynomials are applied to construct basis function [4]. [PERSON] (2012) reported a work in his doctorate thesis that focuses on a discontinuous Galerkin method for the solution of the Vlasov equation. In finite element space, the basis functions used for representation of the distribution function in the Vlasov equation are allowed to be discontinuous at the cell interface [5]. [PERSON] et al. (2012) applied penalty function in approximation of discrete distribution function. The rate of convergence is greatly improved [6]. For better accuracy in modelling non-equilibrium distribution function, many researchers constructed spectral methods to solve plasma Vlasov equation. [PERSON] et al. (1998) used the Fourier-Hermite polynomial to construct basis functions. Fourth order Runge-Kutta method (RK) is chosen to improve the convergence rate [7]. [PERSON] et al. (2006) constructed a spectral method based on generalized Hermite functions [8]. [PERSON] et al. (2009) proposed a forward semi-Lagrangian spectral method with B-splines basis. Different from previous method, numerical time step can be much larger than particle collision time [9]. Since it is hard to control the truncation error in approximation of gas distribution function, conservative methods based on [PERSON]'s scheme are proposed [10]. [PERSON] (1999) et al. first introduced a semi-Lagrangian conservative method [11]. [PERSON] et al. (2001) constructed a particle based conservative method [12]. Since the accuracy of methods is severely influenced by the phase and amplitude error because of dissipation from the way of reconstruction and the use of slope corrector, [PERSON] et al. (2007) constructed a conservative method using the Hermite spline interpolation [13]. [PERSON] et al. (2011) proposed a finite volume method (FVM) to solve the Vlasov equation, in which the PDE is integrated in discrete phase space and the numerical flux of the distribution function through a cell interface is evaluated using high-order approximations of the cell-face average [14]. [PERSON] et al.