Fundamental form of the electrostatic $\delta f$-PIC algorithm and discovery of a converged numerical instability

TL;DR

This paper analyzes the fundamental form of the electrostatic δf-PIC algorithm, revealing a converged numerical instability and proposing mitigation strategies, thereby advancing the understanding of its stability properties in plasma simulations.

Contribution

It provides the exact analytic form of the δf-PIC algorithm and identifies a fundamental numerical instability, along with mitigation methods, enhancing its reliability in plasma physics simulations.

Findings

Discovered a fully-resolved numerical instability in the δf-PIC algorithm.

Identified an under-resolved instability requiring many particles to stabilize.

Proposed mitigation schemes for the identified instabilities.

Abstract

The $\delta f$ particle-in-cell algorithm has been a useful tool in studying the physics of plasmas, particularly turbulent magnetized plasmas in the context of gyrokinetics. The reduction in noise due to not having to resolve the full distribution function indicates an efficiency advantage over standard ("full-$f$") particle-in-cell. Despite its successes, the algorithm behaves strangely in some circumstances. In this work, we document a fully-resolved numerical instability that occurs in the simplest of multiple-species test cases: the electrostatic $\Omega_H$ mode. There is also a poorly-understood numerical instability that occurs when one is under-resolved in particle number, which may require a prohibitively large number of particles to stabilize. Both of these are independent of the time-stepping scheme, and we conclude that they exist if the time advancement were exact. The exact analytic form of the algorithm is presented, and several schemes for mitigating these instabilities are presented.