Numerical stability of relativistic beam multidimensional PIC simulations employing the Esirkepov algorithm

TL;DR

This paper analyzes numerical instabilities in multidimensional relativistic PIC simulations, deriving a dispersion relation and proposing an alternative interpolation method to minimize these instabilities.

Contribution

It derives a dispersion relation for numerical instabilities in relativistic PIC simulations and introduces an alternative interpolation algorithm to significantly reduce instabilities.

Findings

Existence of specific time steps minimizing instabilities

Good agreement with previous WARP code results

Proposed interpolation method nearly eliminates instabilities at certain time steps

Abstract

Rapidly growing numerical instabilities routinely occur in multidimensional particle-in-cell computer simulations of plasma-based particle accelerators, astrophysical phenomena, and relativistic charged particle beams. Reducing instability growth to acceptable levels has necessitated higher resolution grids, high-order field solvers, current filtering, etc. except for certain ratios of the time step to the axial cell size, for which numerical growth rates and saturation levels are reduced substantially. This paper derives and solves the cold beam dispersion relation for numerical instabilities in multidimensional, relativistic, electromagnetic particle-in-cell programs employing either the standard or the Cole-Karkkainnen finite difference field solver on a staggered mesh and the common Esirkepov current-gathering algorithm. Good overall agreement is achieved with previously reported results of the WARP code. In particular, the existence of select time steps for which instabilities are minimized is explained. Additionally, an alternative field interpolation algorithm is proposed for which instabilities are almost completely eliminated for a particular time step in ultra-relativistic simulations.