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

Abstract

We change a previous time-stepping algorithm for solving a multi-scale Vlasov-Poisson system within a Particle-In-Cell method, in order to do accurate long time simulations. As an exponential integrator, the new scheme allows to use large time steps compared to the size of oscillations in the solution.

Long Time Behaviour of an Exponential Integrator for a Vlasov-Poisson System with Strong Magnetic Field

“…Thus, we can observe that the smaller the distance to the slow manifold is, the smaller the corresponding error is. The reason is that, the closer to the slow manifold a particle is, the smaller the amplitude of its oscillation is, and thus, the smaller the propagation of the error E P through E M is (see [8] for similar comments for a simpler Vlasov model). Then, in long time simulations, the errors are obviously signicant, due to our simple linear approximation of the macroscopic time evolution.…”

“…Next, our aim is to continue rst with the case of a slowly varying magnetic eld. Such a situation will lead to dierent fast periods for dierent particles and thus to adapt our algorithm to be able to handle dierent periods (such a procedure was recently successfully implemented in [8] for a two dimensional Vlasov model). Then, towards the six dimensional case, we need to optimize the implementation of our Particle-In-Cell method in order to produce such a numerical simulation.…”

Adaptive multiresolution semi-Lagrangian discontinuous Galerkin methods for the Vlasov equations

“…This part is devoted to experiment considering the following formulation of Vlasov equation from [25] and [24]…”

A parareal algorithm for a highly oscillating Vlasov-Poisson system with reduced models for the coarse solving

“…The coupled Vlasov-Poisson equations ( 1)-( 2)-(3) have been extensively studied theoretically as well as numerically (see for example [8,10,21,5,6,9]). The difficulty arising in the numerical solving of this system is that the solution displays high oscillations in time when the parameter ε is small.…”

“…This approach gives excellent results for long time simulations where the particle beam is shorter than the case considered in the present contribution. Another method, based on an exponential time integrator of the characteristics of the Vlasov equation, is proposed in [9] (see also the references therein). However, this last paper tackles only the case of equations ( 1)-( 2)-(3) with H = 0 and with a particle beam which is short, as in references [5,6].…”