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

F. Assous, F. Tsipis

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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.

Original languageEnglish
Pages (from-to)136-146
Number of pages11
JournalJournal of Computational and Applied Mathematics
Issue number1
StatePublished - 1 May 2009


  • Error estimates
  • Numerical schemes
  • Paraxial approximation
  • Relativistic beams
  • Vlasov-Maxwell
  • Vlasov-Poisson


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