The geometric theory of charge conservation in particle-in-cell simulations

TL;DR

This paper provides a rigorous theoretical foundation for charge conservation in gauge-symmetric particle-in-cell methods, introducing gauge-compatible splitting algorithms that exactly preserve charge and are symplectic.

Contribution

It develops a comprehensive theoretical framework linking gauge symmetry to charge conservation and introduces a new class of gauge-compatible splitting PIC algorithms.

Findings

Gauge symmetry leads to local charge conservation via Noether's theorem.

Gauge-compatible splitting methods exactly preserve the momentum map and charge.

The proposed explicit symplectic gauge-compatible PIC method ensures exact charge conservation.

Abstract

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods' gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether's second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms -- gauge-compatible splitting methods -- that exactly preserve the momentum map associated with a Hamiltonian system's gauge symmetry -- even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.