Numerical approximation of 3D particle beams by multi-scale paraxial Vlasov-Maxwell equations

F. Assous, Y. Furman

Research output: Contribution to journalArticlepeer-review


Even now, solving numerically the 3D time-dependent Vlasov-Maxwell equations is a challenging problem. Hence, it is always interesting to develop simpler but accurate approximate models. By introducing a small parameter, we derived paraxial asymptotic models that approximate these equations, allowing us to treat relativistic cases, much slower beams or even non-relativistic cases. These models are static or quasi-static and with a nth order accuracy that may be chosen as required. Here, we propose a 3D numerical approximation of this multi-scale model. It is based, for the paraxial Maxwell model, on a finite element method coupled with a Particle-In-Cell approximation of the paraxial Vlasov model. Numerical results illustrated the efficiency of the method.

Original languageEnglish
Article number112186
JournalJournal of Computational Physics
StatePublished - 1 Sep 2023
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.


The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Franck Assous reports financial support was provided by Ariel University. Franck Assous reports a relationship with Ariel University that includes: employment.

FundersFunder number
Ariel University


    • Asymptotic methods
    • Paraxial approximation
    • Particle-in-cell numerical schemes
    • Vlasov-Maxwell equations


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