Application of Lie transform techniques for simulation of a charged particle beam

Abstract

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.

Long-time simulation of a highly oscillatory Vlasov equation with an exponential integrator

“…When the solution is periodic, say y(t n + 2πε) = y n , the second step of the algorithm reduces to an exact statement. Otherwise, it can happen that y(t) 2 = R(t) 2 + V (t) 2 vary slowly in time, typically when the electric field is given by the Poisson equation (see [7]). In this case, the scheme allows to capture the small variations of the amplitude of y since it solves them during the first step.…”

“…When the electric field E ε is zero the beam rotate around the origin with period 2πε. Otherwise, the dynamical system (1.4) can be viewed as a perturbation of the system obtained when the electric field is zero (see [7]). When the electric field E ε is obtained by solving the Poisson equation or if it is given by (1.3), the beam evolves by rotating around the origin in the phase space, and in long times a bunch forms around the center of the beam from which filaments of particles are going out.…”

Homogenization-based numerical methods

“…Introduction. Since the previous opus : "Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence" of DCDS-S (Volume 8, Number 1, February 2015, [37], [36], [5], [50] [48] [1], [4], [12], [16], [41] and [30]), the advance on the topic has been important. It was time to take stock.…”