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In this paper, we propose a moment method to numerically solve the Vlasovequations using the framework of the NRxx method developed in [6, 8, 7] for theBoltzmann equation. Due to the same convection term of the Boltzmann equationand the Vlasov equation, it is very convenient to use the moment expansion inthe NRxx method to approximate the distribution function in the Vlasovequations. The moment closure recently presented in [5] is applied to achievethe globally hyperbolicity so that the local well-posedness of the momentsystem is attained. This makes our simulations using high order momentexpansion accessible in the case of the distribution far away from theequilibrium which appears very often in the solution of the Vlasov equations.With the moment expansion of the distribution function, the acceleration in thevelocity space results in an ordinary differential system of the macroscopicvelocity, thus is easy to be handled. The numerical method we developed cankeep both the mass and the momentum conserved. We carry out the simulations ofboth the Vlasov-Poisson equations and the Vlasov-Poisson-BGK equations to studythe linear Landau damping. The numerical convergence is exhibited in terms ofthe moment number and the spatial grid size, respectively. The variation ofdiscretized energy as well as the dependence of the recurrence time on momentorder is investigated. The linear Landau damping is well captured for differentwave numbers and collision frequencies. We find that the Landau damping ratelinearly and monotonically converges in the spatial grid size. The results arein perfect agreement with the theoretic data in the collisionless case.

Solving Vlasov Equations Using NRxx Method