Rapidly growing numerical instabilities routinely occur in multidimensionalparticle-in-cell computer simulations of plasma-based particle accelerators,astrophysical phenomena, and relativistic charged particle beams. Reducinginstability growth to acceptable levels has necessitated higher resolutiongrids, high-order field solvers, current filtering, etc. except for certainratios of the time step to the axial cell size, for which numerical growthrates and saturation levels are reduced substantially. This paper derives andsolves the cold beam dispersion relation for numerical instabilities inmultidimensional, relativistic, electromagnetic particle-in-cell programsemploying either the standard or the Cole-Karkkainnen finite difference fieldsolver on a staggered mesh and the common Esirkepov current-gatheringalgorithm. Good overall agreement is achieved with previously reported resultsof the WARP code. In particular, the existence of select time steps for whichinstabilities are minimized is explained. Additionally, an alternative fieldinterpolation algorithm is proposed for which instabilities are almostcompletely eliminated for a particular time step in ultra-relativisticsimulations.
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