Numerical stability analysis of the pseudo-spectral analytical time-domain PIC algorithm

Abstract

The pseudo-spectral analytical time-domain (PSATD) particle-in-cell (PIC) algorithm solves the vacuum Maxwell's equations exactly, has no Courant timestep 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.

Enabling Lorentz boosted frame particle-in-cell simulations of laser wakefield acceleration in quasi-3D geometry

“…The NCI arises when a plasma drifts relativistically on the grid. There has been much recent progress in identifying the NCI as the source of the instability, in deriving the numerical dispersion relations and determining growth rates, and in identifying mitigation strategies [11,12,13,14,15,16,17,18]. In Ref.…”

“…Within these schemes, Gauss' Law is satisfied by either directly solving it (as is the case in UPIC [24,25,26]), or by using a current that satisfies the continuity equation through a correction (compensation) to match the current deposition scheme with the Maxwell solver (as is the case in here and in [18]). In the approaches which aim to eliminate the NCI [18,19] (rather than reducing its growth rate for any time step size to that for an optimum time step [15]), the use of an FFT is inevitable (at least in one direction) since it is very challenging to design a current deposition scheme in real space to match the EM dispersion of a modified Maxwell solver for an arbitrary order finite difference scheme.…”

Summary report of working group 2: Computations for accelerator physics

“…Recognizing that the Particle-In-Cell (PIC) method has been established as a credible tool for the modeling of advanced accelerators, it is noted that the field of advanced accelerator concepts has been actively stimulating new research in PIC methodology. Thanks to recent work, the analysis of numerical stability of the PIC methods is now well in hand, has been thoroughly validated by PIC simulations and encompasses many algorithms: finite-difference time-domain (FDTD) [1,2], pseudo-spectral time-domain (PSTD) [3,4], pseudo-spectral analytic time-domain (PSATD) [3] (see Fig. 1-left), and pseudo-spectral arbitrary order time-domain (PSAOTD) [5], with sub-cycling [6].…”

Modeling of laser wakefield acceleration in Lorentz boosted frame using a Quasi-3D OSIRIS algorithm

“…This is seen by the fact that in a frame where the plasma drifts there is a violent numerical instability, called the Numerical Cerenkov Instability (NCI) [15]. The potential of the Lorentz boosted frame technique has led to a detailed reexamination of the NCI and to techniques to mitigate (effectively eliminate) it [16,17,18,19,20,21].…”