A paraxial asymptotic model for the coupled Vlasov‐Maxwell problem in electromagnetics

The time‐dependent Vlasov–Maxwell equations are one of the most complete mathematical equations that model charged particle beams or plasma physics problems. However, the numerical solution of this system often requires a large computational effort. It is worthwhile, whenever possible, to take into account the geometrical or physical particularities of the problem to derive asymptotic simpler approximate models, leading to cheaper simulations. In this paper, we consider the case of high energy short beams, as for example the transport of a bunch of highly relativistic charged particles in the interior of a perfectly conducting hollow tube. We then derive and analyze a new paraxial asymptotic model, that approximates the Vlasov–Maxwell equations and is fourth order accurate with respect to a small parameter $\eta$ which reflects the physical characteristics of the problem. This approach promises to be very powerful in its ability to get an accurate and fast algorithm, easy to be developed.