A Particle-In-Cell Method for Plasmas with a Generalized Momentum Formulation, Part III: A Family of Gauge Conserving Methods

TL;DR

This paper develops a family of gauge-conserving particle-in-cell methods using potential formulations of Maxwell's equations, ensuring charge and magnetic flux conservation through time-discrete gauge conditions.

Contribution

It introduces a new family of spatially co-located field solvers that preserve gauge invariance and charge conservation in particle-in-cell simulations, extending previous work with a focus on time discretization.

Findings

Methods satisfy gauge conditions automatically imply Gauss's law for electricity.

The approach is validated on relativistic Weibel instability and electron cloud simulations.

Spectral spatial discretization combined with various time marching schemes ensures stability and accuracy.

Abstract

In this paper, we introduce a new family of spatially co-located field solvers for particle-in-cell applications which evolve the potential formulation of Maxwell's equations under the Lorenz gauge. Our recent work introduced the concept of time-consistency, which connects charge conservation to the preservation of the gauge at the semi-discrete level. It will be shown that there exists a large family of time discretizations which satisfy this property. Additionally, it will be further shown that for large classes of time marching methods, the satisfaction of the gauge condition automatically implies the satisfaction of Gauss's law for electricity, with the potential formulation ensuring that that Gauss's law for magnetism is satisfied by definition. We focus on popular time marching methods including centered differences, backward differences, and diagonally-implicit Runge-Kutta methods, which are coupled to a spectral discretization in space. We demonstrate the theory by testing the methods on a relativistic Weibel instability and a drifting cloud of electrons.