Solving Vlasov Equations Using NR$xx$ Method

Abstract

In this paper, we propose a moment method to numerically solve the Vlasov equations using the framework of the NR$xx$ method developed in [Z. Cai and R. Li, SIAM J. Sci. Comput., 32 (2010), pp. 2875--2907; Z. Cai, R. Li, and Y. Wang, J. Sci. Comput., 50 (2012), pp. 103--119] for the Boltzmann equation. Due to the same convection term for the Boltzmann equation and the Vlasov equations, the NR$xx$ method can be naturally applied. The moment closure recently presented in [Z. Cai, Y. Fan, and R. Li, Comm. Math. Sci., 11 (2013), pp. 547--571] is applied to achieve the global hyperbolicity, and thus the high order moment approximation is accessible in the case of strong nonequilibrium, which appears very often in the solution of the Vlasov equations. In the moment equations, the acceleration in the velocity space appears to be a source term in the equations of macroscopic velocity, and thus it is not difficult to handle numerically. The numerical method we developed can keep both the mass and the momentum conserved. We carry out the simulations of both the Vlasov--Poisson equations and the Vlasov--Poisson-BGK equations to study the linear Landau damping. The damping is well captured and the numerical convergence is obvious. In the collisionless case, the damping rates of our prediction match the theoretical data perfectly.