Application of Lie Transform Techniques for simulation of a charged particle beam

We study a Lie Transform method for a charged beam under the action of a radial external electric field. The aim of the Lie transform method that is used here is to construct a change of variable which transforms the 2D kinetic problem into a 1D problem. This reduces the dimensionality of the problem and make it easier to solve numerically. After applying the Lie transform method, we truncate the expression of the characteristics of the Vlasov equation and the expression of the Poisson equation in the Lie coordinate system and we develop a numerical method for solving the truncated model and we study its efficiency for the simulation of long time beam evolution.