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

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 n^th 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.