Variational Framework for Structure-Preserving Electromagnetic Particle-In-Cell Methods

TL;DR

This paper develops a structure-preserving variational framework for electromagnetic Particle-In-Cell methods, ensuring conservation properties and Hamiltonian structure through discrete action principles and finite element discretizations.

Contribution

It introduces a general discrete variational scheme for electromagnetic PIC methods that preserves geometric structures and conservation laws, adaptable to various finite element choices.

Findings

The scheme has a discrete Poisson structure leading to a Hamiltonian system.

It can be extended to momentum-preserving discretizations using discrete interior products.

Applications include spline finite elements and spectral discretizations with Fourier transforms.

Abstract

In this article we apply a discrete action principle for the Vlasov--Maxwell equations in a structure-preserving particle-field discretization framework. In this framework the finite-dimensional electromagnetic potentials and fields are represented in a discrete de Rham sequence involving general finite element spaces, and the particle-field coupling is represented by a set of projection operators that commute with the differential operators. With a minimal number of assumptions which allow for a variety of finite elements and shape functions for the particles, we show that the resulting variational scheme has a general discrete Poisson structure and thus leads to a semi-discrete Hamiltonian system. By introducing discrete interior products we derive a second type of space discretization which is momentum preserving, based on the same finite elements and shape functions. We illustrate our method by applying it to spline finite elements, and to a new spectral discretization where the particle-field coupling relies on discrete Fourier transforms.