arXiv subjects

Alexander S. Glasser

Publications and source records attributed to Alexander S. Glasser.

9 recordsLinked to original sources

arXiv2026

Generalized Yee methods: Scalable symplectic finite element Maxwell solvers

Yee's finite-difference method preserves two crucial properties of Maxwell's equations -- locality and symplecticity -- and thereby enjoys two computational advantages: scalability on high-performance architectures and long-time numerical accuracy. In this work, we show that Yee's method is a special case of a class of structure-preserving finite element methods -- termed generalized Yee methods (GYMs) -- that are designed to retain both crucial properties. GYMs are built from de Rham-conforming finite elements and achieve locality through sparse mass matrices and their sparse approximate inverses (SPAIs). We prove that the symplectic structure of GYMs is invariant under such sparse approximations, freeing the choice of sparsification strategy. We introduce a novel sparsification strategy, SPAI-OP, which concentrates accuracy at prescribed wave modes by operator probing. We further extend GYMs to structure-preserving electromagnetic particle-in-cell (PIC) methods, whose symplecticity over particle trajectories requires the smooth fields afforded by higher-order finite elements. GYMs therefore retain the computational virtues of Yee's method while enabling unstructured meshes, higher-order accuracy, spectral adaptivity, and symplectic particle coupling.

math.NA↗

arXiv2022

Discrete Gravity with Local Lorentz Invariance

A novel structure-preserving algorithm for general relativity in vacuum is derived from a lattice gauge theoretic discretization of the tetradic Palatini action. The resulting model of discrete gravity is demonstrated to preserve local Lorentz invariance and symplectic structure.

gr-qc↗

arXiv2022

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

A particle-in-cell algorithm is derived with a canonical Poisson structure in the formalism of finite element exterior calculus. The resulting method belongs to the class of gauge-compatible splitting algorithms, which exactly preserve gauge symmetries and their associated conservation laws via the momentum map. We numerically demonstrate this time invariance of the momentum map and its usefulness in establishing precise initial conditions with a desired initial electric field and fixed background charge. The restriction of this canonical, finite element Poisson structure to the 1X2P phase space is also considered and simulated numerically.

physics.plasm-ph↗

arXiv2020

Spontaneous and explicit parity-time-symmetry breaking in drift wave instabilities

A method of Parity-Time (PT)-symmetry analysis is introduced to study the high dimensional, complicated parameter space of drift wave instabilities. We show that spontaneous PT-symmetry breaking leads to the Ion Temperature Gradient (ITG) instability of drift waves, and the collisional instability is the result of explicit PT-symmetry breaking. A new unstable drift wave induced by finite collisionality is identified. It is also found that gradients of ion temperature and density can destabilize the ion cyclotron waves when PT symmetry is explicitly broken by a finite collisionality.

physics.plasm-ph↗

arXiv2020

$\mathbb{1}$-Loop Theory

A new formalism for lattice gauge theory is developed that preserves Poincaré symmetry in a discrete universe. We define the $\mathbb{1}$-loop, a generalization of the Wilson loop that reformulates classical differential equations of motion as identity-valued multiplicative loops of Lie group elements of the form ${[g_1\cdots g_n]=\mathbb{1}}$. A lattice Poincaré gauge theory of gravity is thus derived that employs a novel matter field construction and recovers Einstein's vacuum equations in the appropriate limit.

hep-th↗

arXiv2020

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

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.

physics.plasm-ph↗

arXiv2019

Lifting Spacetime's Poincaré Symmetries

In the following work, we pedagogically develop 5-vector theory, an evolution of scalar field theory that provides a stepping stone toward a Poincaré-invariant lattice gauge theory. Defining a continuous flat background via the four-dimensional Cartesian coordinates $\{x^a\}$, we `lift' the generators of the Poincaré group so that they transform only the fields existing upon $\{x^a\}$, and do not transform the background $\{x^a\}$ itself. To facilitate this effort, we develop a non-unitary particle representation of the Poincaré group, replacing the classical scalar field with a 5-vector matter field. We further augment the vierbein into a new $5\times5$ fünfbein, which `solders' the 5-vector field to $\{x^a\}$. In so doing, we form a new intuition for the Poincaré symmetries of scalar field theory. This effort recasts `spacetime data', stored in the derivatives of the scalar field, as `matter field data', stored in the 5-vector field itself. We discuss the physical implications of this `Poincaré lift', including the readmittance of an absolute reference frame into relativistic field theory. In a companion paper, we demonstrate that this theoretical development, here construed in a continuous universe, enables the description of a discrete universe that preserves the 10 infinitesimal Poincaré symmetries and their conservation laws.

physics.gen-ph↗

arXiv2019

Restoring Poincaré Symmetry to the Lattice

The following work demonstrates the viability of Poincaré symmetry in a discrete universe. We develop the technology of the discrete principal Poincaré bundle to describe the pairing of (1) a hypercubic lattice `base manifold' labeled by integer vertices-denoted $\{\mathbf{n}\}=\{(n_t,n_x,n_y,n_z)\}$-with (2) a Poincaré structure group. We develop lattice 5-vector theory, which describes a non-unitary representation of the Poincaré group whose dynamics and gauge transformations on the lattice closely resemble those of a scalar field in spacetime. We demonstrate that such a theory generates discrete dynamics with the complete infinitesimal symmetry-and associated invariants-of the Poincaré group. Following our companion paper, we `lift' the Poincaré gauge symmetries to act only on vertical matter and solder fields, and recast `spacetime data'--stored in the $\partial_μϕ(x)$ kinetic terms of a free scalar field theory--as `matter field data'-stored in the $ϕ^μ[\mathbf{n}]$ components of the 5-vector field itself. We gauge 5-vector theory to describe a lattice gauge theory of gravity, and discuss the physical implications of a discrete, Poincaré-invariant theory.

physics.gen-ph↗

arXiv2018

Kelvin-Helmholtz instability is the result of parity-time symmetry breaking

Parity-Time (PT)-symmetry is being actively investigated as a fundamental property of observables in quantum physics. We show that the governing equations of the classical two-fluid interaction and the incompressible fluid system are PT-symmetric, and the well-known Kelvin-Helmholtz instability is the result of spontaneous PT-symmetry breaking. It is expected that all classical conservative systems governed by Newton's law admit PT-symmetry, and the spontaneous breaking thereof is a generic mechanism for classical instabilities. Discovering the PT-symmetry of systems in fluid dynamics and plasma physics and identifying the PT-symmetry breaking responsible for instabilities enable new techniques to classical physics and enrich the physics of PT-symmetry.

physics.flu-dyn↗