Parametric instabilities in cold drifting-beam PIC discretizations

Characterize whether the time-periodic dynamical matrices induced by grid aliasing in momentum-conserving and energy-conserving particle-in-cell discretizations of a cold drifting beam produce parametric instabilities, and determine the associated resonance conditions.

Background

For a cold beam with velocity v_0, the particle lattice drifts through the fixed computational grid. Because the grid-based force evaluation contains aliasing errors, the linearized dynamical matrix becomes periodic in time with fundamental frequency Ω = k_g v_0. The paper relates this periodic coefficient matrix to a parametrically driven oscillator and identifies the fundamental Mathieu resonance as the case in which Ω is twice the natural frequency.

The authors do not resolve the resulting stability problem for either the momentum-conserving or energy-conserving formulation in this paper; they state that the issue will be treated in a subsequent publication.

References

This oscillation in time of the matrix elements of M at frequency Ω=k_gv_0 for v_0≠0 leads to the possibility of parametric instabilities, for both the MCP and ECP forms.

— Analysis of grid instabilities in particle-in-cell codes based on a meshfree approach  (2610.02052 - Finn et al., 1 Oct 2026) in Section 6, “Cold beam v_0 ≠ 0”