Time integrator agnostic charge conserving finite element PIC

Abstract

Developing particle-in-cell (PIC) methods using finite element basis sets, and without auxiliary divergence cleaning methods, was a long-standing problem until recently. It was shown that if consistent spatial basis functions are used, one can indeed create a methodology that was charge conserving, albeit using a leapfrog time stepping method. While this is a significant advance, leapfrog schemes are only conditionally stable and time step sizes are closely tied to the underlying mesh. Ideally, to take full advantage of advances in finite element methods (FEMs), one needs a charge conserving PIC methodology that is agnostic to the time stepping method. This is the principal contribution of this paper. In what follows, we shall develop this methodology, prove that both charge and Gauss' laws are discretely satisfied at every time step, provide the necessary details to implement this methodology for both the wave equation FEM and Maxwell solver FEM, and finally demonstrate its efficacy on a suite of test problems. The method will be demonstrated by single particle evolution, non-neutral beams with space-charge, and adiabatic expansion of a neutral plasma, where the Debye length has been resolved, and real mass ratios are used.

An Envelope Tracking Approach for Finite Element Particle-in-Cell Simulations

“…T matrix effects a divergence of the flux density (which lies in the dual grid) in terms of quantities defined on the primal grid. In practice, as one only evolves the curl equations, the solution at every time step will automatically satisfy (12) provided care is taken in constructing the right hand side [13]. But what is critical for satisfaction of Gauss' law is that this discrete solution also satisfy (13).…”

“…In addition, work has been done on constructing a structure-preserving FEMPIC scheme [10], [11] based on Whitney forms defined by B-splines [12]. Furthermore, it is possible to sidestep the Courant-Friedrichs-Lewy (CFL) constraint on the timestep size by using unconditionally stable implicit time marching schemes while conserving charge to machine precision [13]. And finally, techniques have been very recently developed that allow for explicit satisfaction of the Coulomb Gauge, thus alleviating the problem of spurious nullspace solutions that exist in most implicit implementations [14].…”

“…Extending these results to a system with moving macroparticles is challenging for two primary reasons: (1) While the fields themselves oscillate within a narrowband high frequency window, the particle positions and velocities generally have significant DC components that force analysis of Newton's equations at the highest frequency of interest. (2) While recent advances make it possible to conserve charge within an implicit time marching scheme [13], these techniques break down when used with an analytically prescribed plane wave component (We discuss this in greater detail in Section III-B).…”

A gauge-compatible Hamiltonian splitting algorithm for particle-in-cell simulations using finite element exterior calculus

“…Structure-preserving particle-in-cell (PIC) algorithms preserve many of the geometric and topological mathematical structures of a point-particle kinetic plasma model, including its symplectic structure, symmetries, conservation laws and cohomology (Villasenor & Buneman 1992;Esirkepov 2001;Squire, Qin & Tang 2012;Evstatiev & Shadwick 2013;Xiao et al 2013;Crouseilles, Einkemmer & Faou 2015;He et al 2015;Moon, Teixeira & Omelchenko 2015;Qin et al 2015Qin et al , 2016Xiao et al 2015;He et al 2016;Burby 2017;Kraus & Hirvijoki 2017;Morrison 2017;Xiao, Qin & Liu 2018;Xiao & Qin 2019;Glasser & Qin 2020;Hirvijoki, Kormann & Zonta 2020;Holderied, Possanner & Wang 2021;O'Connor et al 2021;Perse, Kormann & Sonnendrücker 2021;Pinto, Kormann & Sonnendrücker 2022;Wang et al 2021;Xiao & Qin 2021). One such structure, gauge symmetry, was first preserved in a PIC algorithm in the Lagrangian formalism via a variational method (Squire et al 2012).…”

“…2017; Morrison 2017; Xiao, Qin & Liu 2018; Xiao & Qin 2019; Glasser & Qin 2020; Hirvijoki, Kormann & Zonta 2020; Holderied, Possanner & Wang 2021; Kormann & Sonnendrücker 2021; O'Connor et al. 2021; Perse, Kormann & Sonnendrücker 2021; Pinto, Kormann & Sonnendrücker 2022; Wang et al. 2021; Xiao & Qin 2021).…”