Fundamental form of the electrostatic δf -PIC algorithm and discovery of a converged numerical instability

The δf 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 the 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 Ω_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 also presented.