  Rapidly growing numerical instabilities routinely occur in multidimensional
particle-in-cell computer simulations of plasma-based particle accelerators,
astrophysical phenomena, and relativistic charged particle beams. Reducing
instability growth to acceptable levels has necessitated higher resolution
grids, high-order field solvers, current filtering, etc. except for certain
ratios of the time step to the axial cell size, for which numerical growth
rates and saturation levels are reduced substantially. This paper derives and
solves the cold beam dispersion relation for numerical instabilities in
multidimensional, relativistic, electromagnetic particle-in-cell programs
employing either the standard or the Cole-Karkkainnen finite difference field
solver on a staggered mesh and the common Esirkepov current-gathering
algorithm. Good overall agreement is achieved with previously reported results
of the WARP code. In particular, the existence of select time steps for which
instabilities are minimized is explained. Additionally, an alternative field
interpolation algorithm is proposed for which instabilities are almost
completely eliminated for a particular time step in ultra-relativistic
simulations.

Plasma Physics;Accelerator Physics