This paper addresses the numerical issues related to the modeling of beams or plasmas crossing at relativistic velocity using the particle-in-cell method. Issues related to the use of the standard Boris particle pusher are identified and a novel pusher which circumvents them is proposed, whose effectiveness is demonstrated on single particle tests. A procedure for solving the fields is proposed, which retains electrostatic, magnetostatic, and inductive field effects in the direction of the mean velocity of the species, is fully explicit and simpler than the full Darwin approximation. Finally, results are given, from a calculation using the novel features, of an ultrarelativistic beam interacting with a background of electrons.

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