Numerical paraxial approximation for highly relativistic beams

The numerical modelling in plasma physics as well as in microwave devices or accelarator technology, requires to develop methods for solving the time-dependent Vlasov-Maxwell problem. Eventhough the numerical approximation of this complex system of equations is necessary in a number of cases, it leads however to very expensive computations. For this reason, simpler models have been developed in many situations. Hence, when dealing with charged particle beams, it is often fruitful to exploit the paraxial property, i.e the property that the particles of the beam remain close to an optical axis. In a previous paper, such a model was developed and mathematically analyzed. We present here the resulting axisymmetric model, in case of highly relativistic beams. Then, numerical schemes are constructed and results to illustrate the efficiency of the model are presented.