  We studied the stability property of numerical Cherenkov radiation in
relativistic plasma flows employing particle-in-cell simulations. Using the
implicit finite-difference time-domain method to solve Maxwell equations, we
found that nonphysical instability was greatly inhibited with a
Courant-Friedrichs-Lewy (CFL) number of 1.0. The present result contrasts with
recently reported results (Vay, J. L., et al. 2011, J. Comp. Phys, 230, 5908;
Godfrey. B., & Vay, J. L. 2013, J. Comp. Phys, 243, 260; Xu, X., et al. 2013,
Comput. Phys. Commun, 184, 2503) in which magical CFL numbers in the range
0.5-0.7 were obtained with explicit field solvers. In addition, we found
employing higher-order shape functions and an optimal implicitness factor
further suppressed long-wavelength modes of the instability. The findings
allowed the examination of the long-term evolution of a relativistic
collisionless shock without the generation of nonphysical wave excitations in
the upstream. This achievement will allow us to investigate particle
accelerations in relativistic shocks associated with, for example, gamma-ray
bursts.
