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

Abstract

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 | 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.

Modeling of laser wakefield acceleration in Lorentz boosted frame using EM-PIC code with spectral solver

“…By carefully choosing the Maxwell solver, the instability pattern can be manipulated so that mitigation can be achieved. As discussed in [12,14,15,17], when a spectral solver is used, there are no intersections of the EM modes with the main beam resonance. As a result, the instability can be found only at the aliased resonances and the fatest growing modes are the first spatial aliases.…”

“…The numerical Cerenkov instability induced by relativistic plasma drift has been extensively studied in [16,17]. In a PIC system, when the plasma is drifting relativistically, the velocity of the drifting particles can be equal (be in resonance) to the component of the phase velocity of the main EM mode along the drift direction.…”

“…However, in the boosted frame LWFA simulations noise from a numerical instability can be an issue. As discussed in [12,13,14,15,16,17], the noise results from a numerical Cerenkov instability induced by the plasma drifting with relativistic speeds on the grid. According to the dispersion relation this numerical instability is attributed to the coupling between the wave-particle resonances with EM modes (including aliased modes) in the numerical system.…”

“…[16,17], when using the FDTD code to simulate relativistic plasma drift, an optimized time step has to be chosen to minimize the instability growth rate. While the instability growth rate is minimized, this time step does not lead to complete elimination of the instability and it can lead to further errors in numerical dispersion.…”

“…According to the dispersion relation this numerical instability is attributed to the coupling between the wave-particle resonances with EM modes (including aliased modes) in the numerical system. The pattern of the instability in Fourier space can be found at the intersections of the EM dispersion relation of the solver used in the simulation algorithm, and the wave-particle resonances [15,16,17].…”