Perfect Conductor Boundary Conditions for Geometric Particle-in-Cell Simulations of the Vlasov-Maxwell System in Curvilinear Coordinates

TL;DR

This paper extends geometric electromagnetic particle-in-cell methods to curvilinear coordinates with perfect conductor boundary conditions, ensuring conservation and stability in simulations of the Vlasov-Maxwell system.

Contribution

It introduces a de Rham sequence with spline functions for boundary conditions and develops semi-explicit and semi-implicit schemes that conserve energy and Gauss's law.

Findings

Achieved stable long-term simulations with conservation properties.

Developed efficient preconditioning for spline finite element mass matrices.

Extended GEMPIC to realistic boundary conditions in curvilinear coordinates.

Abstract

Structure-preserving methods can be derived for the Vlasov-Maxwell system from a discretisation of the Poisson bracket with compatible finite-elements for the fields and a particle representation of the distribution function. These geometric electromagnetic particle-in-cell (GEMPIC) discretisations feature excellent conservation properties and long-time numerical stability. This paper extends the GEMPIC formulation in curvilinear coordinates to realistic boundary conditions. We build a de Rham sequence based on spline functions with clamped boundaries and apply perfect conductor boundary conditions for the fields and reflecting boundary conditions for the particles. The spatial semi-discretisation forms a discrete Poisson system. Time discretisation is either done by Hamiltonian splitting yielding a semi-explicit Gauss conserving scheme or by a discrete gradient scheme applied to a Poisson splitting yielding a semi-implicit energy-conserving scheme. Our system requires the inversion of the spline finite element mass matrices, which we precondition with the combination of a Jacobi preconditioner and the spectrum of the mass matrices on a periodic tensor product grid.