A PIC method for solving a paraxial model of highly relativistic beams

Solving the Vlasov-Maxwell problem can lead to very expensive computations. To construct a simpler model, Laval et al. [G. Laval, S. Mas-Gallic, P.A. Raviart, Paraxial approximation of ultrarelativistic intense beams, Numer. Math. 69 (1) (1994) 33-60] proposed to exploit the paraxial property of the charged particle beams, i.e the property that the particles of the beam remain close to an optical axis. They so constructed a paraxial model and performed its mathematical analysis. In this paper, we investigate how their framework can be adapted to handle the axisymmetric geometry, and its coupling with the Vlasov equation. First, one constructs numerical schemes and error estimates results for this discretization are reported. Then, a Particle In Cell (PIC) method, in the case of highly relativistic beams is proposed. Finally, numerical results are given. In particular, numerical comparisons with the Vlasov-Poisson model illustrate the possibilities of this approach.