Numerical stability analysis of the Pseudo-Spectral Analytical Time-Domain PIC algorithm

TL;DR

This paper analyzes the numerical stability of the PSATD particle-in-cell algorithm, showing it generally offers superior stability and can be made nearly free of numerical Cherenkov instability with filtering, especially for relativistic simulations.

Contribution

It derives and compares the numerical dispersion relation of PSATD with PSTD and FDTD, demonstrating improved stability and stability enhancements with filtering.

Findings

PSATD has superior stability over a range of time steps.

Digital filtering significantly reduces numerical Cherenkov instability.

PSATD can be nearly free of instability for relativistic simulations with proper filtering.

Abstract

The pseudo-spectral analytical time-domain (PSATD) particle-in-cell (PIC) algorithm solves the vacuum Maxwell's equations exactly, has no Courant time-step limit (as conventionally defined), and offers substantial flexibility in plasma and particle beam simulations. It is, however, not free of the usual numerical instabilities, including the numerical Cherenkov instability, when applied to relativistic beam simulations. This paper derives and solves the numerical dispersion relation for the PSATD algorithm and compares the results with corresponding behavior of the more conventional pseudo-spectral time-domain (PSTD) and finite difference time-domain (FDTD) algorithms. In general, PSATD offers superior stability properties over a reasonable range of time steps. More importantly, one version of the PSATD algorithm, when combined with digital filtering, is almost completely free of the numerical Cherenkov instability for time steps (scaled to the speed of light) comparable to or smaller than the axial cell size.