S. V. Zaharchenko, A. G. Shein

Studying dynamics of high-current long relativistic electron beams helps develop new microwave devices with high power output. Known limits exist for current densities in compensated and uncompensated beams. The widely-used particle-in-cell (PIC) meth-ods for modeling of electron beams map the simulation space to a cellular mesh, which may lead to inaccuracies in the beam dynamics. Particle-to-particle methods (PPM) are not so widely used because of lower performance, but they can provide finer details of the beam-s structure. Acceptable performance can be achieved by using parallel computations. It is customary to model electron beams using beams of "large" particles, for which the equation of motions are solved together with the equations for the electromagnetic field. In the PPM method the field in the relativistic case is calculated using the Lienard-Weichert formulae. We show that in the relativistic case motion equations should be solved by integrating force to get momentum, instead of integrating acceleration to get velocity. In a strong magnetic field it is important to make sure the field does not influence a particle-s longitudinal velocity and only changes the direction of the velocity. Using conventional linear multi-step methods for motion equation integration may lead to superluminal average velocities even though instant particle velocities are bounded. However, this can be detected and corrected using a lower-order method for a particular step. A different solution which enables better total energy conservation is using the Verlet leapfrog integrator. Results of numeric experiments using the methods described are presented. The model can describe the position and time of emergence of a virtual cathode. It is shown that beams can partially pass through the virtual cathode.