Numerical instability due to relativistic plasma drift in EM-PIC simulations

The numerical instability observed in the Electromagnetic-Particle-in-cell (EM-PIC) simulations with a plasma drifting with relativistic velocities is studied using both theory and computer simulations. We derive the numerical dispersion relation for a cold plasma drifting with a relativistic velocity and find an instability attributed to the coupling between the beam modes of the drifting plasma and the electromagnetic modes in the system. The characteristic pattern of the instability in Fourier space for various simulation setups and Maxwell Equation solvers are explored by solving the corresponding numerical dispersion relations. Furthermore, based upon these characteristic patterns we derive an asymptotic expression for the instability growth rate. The asymptotic expression greatly speeds up the calculation of instability growth rate and makes the parameter scan for minimal growth rate feasible even for full three dimensions. The results are compared against simulation results and good agreement is found. These results can be used as a guide to develop possible approaches to mitigate the instability. We examine the use of a spectral solver and show that such a solver when combined with a low pass filter with a cutoff value of $|\vec{k}|$ essentially eliminates the instability while not modifying modes of physical interest. The use of spectral solver also provides minimal errors to electromagnetic modes in the lowest Brillouin zones.