Collisional effects on the numerical recurrence in Vlasov-Poisson simulations

Abstract

The initial state recurrence in numerical simulations of the Vlasov-Poisson system is a well-known phenomenon. Here we study the effect on recurrence of artificial collisions modeled through the Lenard-Bernstein operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456-1459(1958]. By decomposing the linear Vlasov-Poisson system in the Fourier-Hermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bump-on-tail instability. The analysis is then confirmed and extended to the nonlinear regime through a Eulerian collisional Vlasov-Poisson code. It is found that, despite being routinely used, an artificial collisionality is not a viable way of preventing recurrence in numerical simulations without compromising the kinetic nature of the solution. Moreover, it is shown how numerical effects associated to the generation of fine velocity scales, can modify the physical features of the system evolution even in nonlinear regime. This means that filamentation-like phenomena, usually associated with low amplitude fluctuations contexts, can play a role even in nonlinear regime.

Solar wind collisional heating

“…Although this approach is quite ‘unphysical’ (collisions naturally act in three dimensions), these operators can satisfactorily model collisions in laboratory plasmas devices, such as the Penning–Malmberg traps, where the plasma is confined into a long and thin column and the dynamics occurs mainly along a single direction (Anderson & O’Neil 2007 a , b ; Pezzi et al. 2013; Pezzi, Camporeale & Valentini 2016 b ).…”

Laboratory space physics: Investigating the physics of space plasmas in the laboratory

“…468 To understand how energy is cascaded by turbulence through the six-dimensional phase-space of weakly collisional heliospheric plasmas, recent studies have employed a Hermite spectral representation of the structures in velocity space. [469][470][471][472][473][474][475][476][477][478][479][480][481] Such an optimal spectral representation of the deviations from equilibrium in the particle velocity distribution functions has lead to the discovery of an unanticipated process, called anti-phase-mixing, that may inhibit collisionless damping in a turbulent environment. 477,478 Such elegant spectral methods maximize the scientific return from the detailed measurements of fluctuations in velocity-space that can be made both in the laboratory and by modern spacecraft instrumentation.…”

Suppression of Recurrence in the Hermite-Spectral Method for Transport Equations

“…In this section, we will generalize the result to the linear Landau damping problem, for which the recurrence in the electric energy has been widely observed [24,10]. To study the linear Landau damping, we consider the dimensionless Vlasov-Poisson (VP) equation which models the motion of a collection of charged particles in the self-consistent electric field.…”

“…The similar idea is adopted in [3] to get a slightly nonlinear discretization. For transform methods, one can suppress recurrence by introducing filters to the numerical methods [23,7], or adding artificial collisions to the model [4,24]. The suppression of the recurrence is numerically analyzed in [16], where it is shown that the collision has a damping effect for the high-frequency modes, so that the distribution function is smoothed out and the filamentation is weakened.…”