Stability and conservation properties of Hermite-based approximations of the Vlasov-Poisson system

TL;DR

This paper develops Hermite spectral methods with artificial collision operators for the Vlasov-Poisson system, ensuring stability and conservation of physical invariants, with analysis for linear and nonlinear cases.

Contribution

It introduces a Hermite spectral approximation with designed collision operators that preserve invariants and provides stability analysis for both linear and nonlinear Vlasov-Poisson models.

Findings

Stability conditions relate collision term magnitude, spectral modes, and time-step.

Conservation of particles, momentum, and energy is maintained.

Analysis extends from linear to nonlinear Vlasov-Poisson models.

Abstract

Spectral approximation based on Hermite-Fourier expansion of the Vlasov-Poisson model for a collisionless plasma in the electro-static limit is provided, by including high-order artificial collision operators of Lenard-Bernstein type. These differential operators are suitably designed in order to preserve the physically-meaningful invariants (number of particles, momentum, energy). In view of time-discretization, stability results in appropriate norms are presented. In this study, necessary conditions link the magnitude of the artificial collision term, the number of spectral modes of the discretization, as well as the time-step. The analysis, carried out in full for the Hermite discretization of a simple linear problem in one-dimension, is then partly extended to cover the complete nonlinear Vlasov-Poisson model.