GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods

TL;DR

This paper introduces a new geometric finite element particle-in-cell framework for the Vlasov-Maxwell system that preserves key physical invariants and maintains stability over long simulations.

Contribution

It develops a Hamiltonian-preserving discretization using finite element exterior calculus, ensuring conservation laws and stability in electromagnetic plasma simulations.

Findings

Conservation of charge and divergence constraints achieved.

Exact energy and momentum preservation over long simulations.

Framework guarantees invariants regardless of finite element basis choice.

Abstract

We present a novel framework for Finite Element Particle-in-Cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the Finite Element basis, as long as the corresponding Finite Element spaces satisfy certain compatibility conditions.