Gauge Structure in Algorithms for Plasma Physics

Abstract

In recent years, structure-preserving algorithms have proliferated in computational plasma physics. Such algorithms rigorously preserve geometric and topological features of the plasmas they model. This dissertation studies a particular feature of these algorithms, namely, their preservation of gauge symmetry and conservation laws—collectively described as their gauge structure. In this work, new extensions of Noether's theorems are discovered in discrete settings, and a comprehensive theoretical grounding for charge conservation is thereby established for gauge-symmetric particle-in-cell (PIC) algorithms for plasma physics. First, a new, general class of algorithms—gauge-compatible splitting methods—is discovered that exactly upholds the Noether principle in discrete Hamiltonian systems. Second, a method of this class is constructed for a PIC algorithm, whose electromagnetic gauge structure is systematically analyzed via the momentum map. Third, this result is extended to a PIC method defined using finite element exterior calculus, and it is tested numerically therewith. Fourth, a formulation of Noether's second theorem is developed and applied toward a variational PIC algorithm using the formalism of discrete exterior calculus. The successful preservation of electromagnetic gauge structure in discrete algorithms motivates the exploration of a related problem in the appendices of this thesis. In particular, while internal gauge symmetries are demonstrated to be well preserved in structure-preserving simulations of physical systems, their spacetime symmetries are crucially forfeited in algorithms that discretize the spacetime manifold. In plasma physics algorithms, the breaking of spacetime symmetry generally leads to an unfortunate loss of energy-momentum conservation. While continuous spacetime and its symmetries are incompatible with discretely defined simulations, an alternative realization of these symmetries is more amenable to algorithmic description. In the appendices, a discrete field theory is pursued that reinterprets the Poincaré symmetries of spacetime as internal gauge transformations. New finite-dimensional Poincaré representations are explored as the targets of such gauge transformations, and a method is developed to construct their matter currents. Further applying such a representation, a structure-preserving algorithm for gravitational simulations in vacuum is also derived from a novel discretization of the tetradic Palatini action.