A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations

Franck Assous,Yevgeni Furman

doi:10.5802/crmeca.50

Abstract

We introduce a new family of paraxial asymptotic models that approximate the Vlasov-Maxwell equations in non-relativistic cases. This formulation is $n$-th order accurate in a parameter $\eta$, which denotes the ratio between the characteristic velocity of the beam and the speed of light. This family of models is interesting, first because it is simpler than the complete Vlasov-Maxwell equation, then because it allows us to choose the model complexity according to the expected accuracy.

Highlights

Charged particle beams are very useful in a variety of scientific and technological applications
In the case of high-energy, ultra-relativistic short beams, Laval et al [14] derived a paraxial approximation of the Vlasov–Maxwell equations by introducing a moving frame, which travels along the optical axis at the speed of light c
We proposed a new family of paraxial asymptotic models that approximate the non-relativistic Vlasov–Maxwell equations

Summary

Introduction

Charged particle beams are very useful in a variety of scientific and technological applications. In the case of high-energy, ultra-relativistic short beams, Laval et al [14] derived a paraxial approximation of the Vlasov–Maxwell equations by introducing a moving frame, which travels along the optical axis at the speed of light c. This idea of changing variables to follow the moving frame is not new; it can be found elsewhere, for instance in [18, 19].

Vlasov–Maxwell model

Scaling of equations

Asymptotic expansion

Paraxial model

Conclusion