Elimination of the numerical Cerenkov instability for spectral EM-PIC codes

TL;DR

This paper investigates and addresses the numerical Cerenkov instability in spectral electromagnetic particle-in-cell codes used for simulating relativistically drifting plasmas, revealing additional unstable modes and proposing solutions to eliminate them.

Contribution

The study extends understanding of NCI by identifying new unstable modes in spectral EM-PIC codes and provides analytic expressions for their growth rates and locations.

Findings

Spectral EM-PIC codes exhibit additional unstable modes when filtering out fastest growing modes.

Unstable modes are linked to coupling between transverse electromagnetic and longitudinal Langmuir modes.

Analytic dispersion relations are derived for predicting mode locations and growth rates.

Abstract

When using an electromagnetic particle-in-cell (EM-PIC) code to simulate a relativistically drifting plasma, a violent numerical instability known as the numerical Cerenkov instability (NCI) occurs. The NCI is due to the unphysical coupling of electromagnetic waves on a grid to wave-particle resonances, including aliased resonances, i.e., $\omega + 2\pi\mu/\Delta t=(k_1+ 2\pi\nu_1/\Delta x_1)v_0$, where $\mu$ and $\nu_1$ refer to the time and space aliases and the plasma is drifting relativistically at velocity $v_0$ in the $\hat{1}$-direction. Recent studies have shown that an EM-PIC code which uses a spectral field solver and a low pass filter can eliminate the fastest growing modes of the NCI. Based on these studies a new spectral PIC code for studying laser wakefield acceleration (LWFA) in the Lorentz boosted frame was developed. However, we show that for parameters of relevance for LWFA simulations in the boosted frame, a relativistically drifting plasma is susceptible to a host of additional unstable modes with lower growth rates, and that these modes appear when the fastest growing unstable modes are filtered out. We show that these modes are most easily identified as the coupling between modes which are purely transverse (EM) and purely longitudinal (Langmuir) in the rest frame of the plasma for specific time and space aliases. We rewrite the dispersion relation of the drifting plasma for a general field solver and obtain analytic expressions for the location and growth rate for each unstable mode, i.e, for each time and space aliased resonances. We show for the spectral solver that when the fastest growing mode is eliminated a new mode at the fundamental resonance ($\mu=\nu_1=0$) can be seen. (Please check the whole abstract in the paper).