Long-time simulation of a highly oscillatory Vlasov equation with an exponential integrator

The simplified model of the full Vlasov-Maxwell system is particularly adapted to the study of long and thin beams of charged particles within the paraxial approximation assuming that the beam is axisymmetric. In this article we change a previous time-stepping algorithm for solving a multi-scale Vlasov-Poisson system within a Particle-In-Cell method, in order to perform accurate long-time simulations. As an exponential integrator, the new scheme allows us to use large time steps compared to the size of the oscillations in the solution