  The pseudo-spectral analytical time-domain (PSATD) particle-in-cell (PIC)
algorithm solves the vacuum Maxwell's equations exactly, has no Courant
time-step limit (as conventionally defined), and offers substantial flexibility
in plasma and particle beam simulations. It is, however, not free of the usual
numerical instabilities, including the numerical Cherenkov instability, when
applied to relativistic beam simulations. This paper presents several
approaches that, when combined with digital filtering, almost completely
eliminate the numerical Cherenkov instability. The paper also investigates the
numerical stability of the PSATD algorithm at low beam energies.
