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The geometric theory of charge conservation in particle-in-cell simulations

Part of: Hamiltonian Methods in Plasma Physics

Published online by Cambridge University Press:  27 May 2020

Alexander S. Glasser Open the ORCID record for Alexander S. Glasser [Opens in a new window]  and
Hong Qin
Alexander S. Glasser*
Affiliation:
Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
Hong Qin
Affiliation:
Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.

Keywords

plasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 3 , June 2020 , 835860303
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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