On a penalization of the Child–Langmuir emission condition for the one-dimensional Vlasov–Poisson equation

Abstract

In a preceding paper, an asymptotic analysis of the one dimensional Vlasov–Poisson was presented. That analysis accounted for the “Child–Langmuir regime”, when the injected particle energy is small compared with the applied external bias. In the present paper, we propose a numerical algorithm for solving the Vlasov-Poisson equations in the “Child–Langmuir regime”. The crucial step of the algorithm is a penalization of the “Child–Langmuir emission condition”, which is proved to converge in the one-dimensional case.