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Abstract: This paper is concerned with the numerical simulation of highly relativistic beams. We developed a particle-in-cell code for highly relativistic beams, based on the paraxial Vlasov–Maxwell formulation of Laval et al. This formulation follows the beam in a speed-of-light frame. It is fourth order accurate in the small characteristic velocity of the beam. The formulation is simpler than standard particle-in-cell codes in the lab frame or in the beam frame and gives a fast and easy to implement algorithm. Numerical examples illustrate the accuracy of the method. [Copyright &y& Elsevier]