Abstract Details
Abstracts
Author: Brad A Shadwick
Requested Type: Consider for Invited
Submitted: 2024-03-29 14:40:00
Co-authors: Adam Higuet
Contact Info:
Univ of Nebraska - Lincoln
208 Jorgensen Hall
Lincoln, NEBRASKA 68588-0
United States
Abstract Text:
Recently there has been significant interest in formulations of macro-particle models using variational methods. This is attractive because many of the inherent pathologies of traditional PIC methods are avoided. Broadly, there are two approaches to constructing these models that on the surface appear incompatible: a Lagrangian method based on the Low Lagrangian and a Hamiltonian method based on the noncanonical Vlasov-Maxwell Poisson bracket. In both cases, a reduction is performed on the distribution function, which is replaced by a finite sum of macro-particles with a fixed spatial structure and definite momentum. The distribution is thus replaced as dynamical variable by a collection of marco-particle positions and momenta. In the Lagrangian formulation, it is natural to represent the fields on a grid. Doing so yields an algorithm that is of the same computational cost as the traditional PIC algorithm but is free of grid-heating and generally has lower noise. It does not appear possible to introduce a grid for the field in the noncanonical formulation. Instead, a basis function expansion has used based on discrete exterior calculus. The basis approach, which can also be used in the Lagrangian setting, has the disadvantage of introducing "mass matrices" resulting in all-to-all communications in a parallel implementation. Here we resolve the apparent discrepancy between the two approaches and show: 1) that the noncanonical Hamiltonian formulation can be transformed into a canonical Hamiltonian system; 2) that this canonical Hamiltonian system is identical to that obtained from the Lagrangian by Legendre transformation; and 3) that careful treatment of the variational principle in both the Lagrangian and Hamiltonian settings is necessary to maintain charge conservation. In the Lagrangian setting, introducing a grid (which is required to be computationally performant) breaks gauge invariance leading to an unusual structure of the variational principle.
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