Exact local conservation of energy in fully implicit PIC algorithms

TL;DR

This paper proves that fully implicit particle-in-cell algorithms strictly conserve local energy at the discrete level, extending previous global conservation results to local flux balance and including orbit averaging and electromagnetic effects.

Contribution

It establishes a local energy conservation theorem for fully implicit PIC algorithms, valid in multiple dimensions and models, including orbit averaging and electromagnetic effects.

Findings

Existence of a local energy conservation theorem for fully implicit PIC algorithms.

The theorem applies to 1D electrostatic, electromagnetic, and orbit-averaged models.

Numerical demonstration confirms the local energy conservation property.

Abstract

We consider the issue of strict, fully discrete \emph{local} energy conservation for a whole class of fully implicit local-charge- and global-energy-conserving particle-in-cell (PIC) algorithms. Earlier studies demonstrated these algorithms feature strict global energy conservation. However, whether a local energy conservation theorem exists (in which the local energy update is governed by a flux balance equation at every mesh cell) for these schemes is unclear. In this study, we show that a local energy conservation theorem indeed exists. We begin our analysis with the 1D electrostatic PIC model without orbit-averaging, and then generalize our conclusions to account for orbit averaging, multiple dimensions, and electromagnetic models (Darwin). In all cases, a temporally, spatially, and particle-discrete local energy conservation theorem is shown to exist, proving that these formulations (as originally proposed in the literature), in addition to being locally charge conserving, are strictly locally energy conserving as well. In contrast to earlier proofs of local conservation in the literature \citep{xiao2017local}, which only considered continuum time, our result is valid for the fully implicit time-discrete version of all models, including important features such as orbit averaging. We demonstrate the local-energy-conservation property numerically with a paradigmatic numerical example.