A well-balanced scheme using exact solutions to the two species Vlasov-Poisson system

Abstract

In this work, we consider the numerical approximation of the two-species Vlasov-Poisson system using Eulerian methods. A family of exact non homogeneous stationary solutions are constructed using elliptic functions. Then, specific numerical schemes are proposed to compute solutions which remain close to a given stationary solution since standard schemes fail to capture such a dynamics on coarse meshes. The strategy is based on a suitable decomposition of the solution (in the spirit of δ f approaches) which can be easily combined with any classical Vlasov solvers. For unstable dynamics, a projection technique is proposed in order to dynamically change the equilibrium. Finally, numerical tests are proposed to illustrate the good behavior of the proposed strategy.