Modeling and Simulation Group Meeting
25 min talks
Speaker: Alexandre Milewski and Paco Rilloraza, Courant Institute, NYU
Location: Warren Weaver Hall 202
Date: Thursday, November 14, 2024, 12:30 p.m.
Synopsis:
Alexandre Milewski: "History of Physics: The Umdeutung paper"
In the 1920's, a hot-button issue in the world of theoretical physics was the modelling of the hydrogen atom. Models of the time essentially consisted of a list of quantization conditions imposed on a classical system in order to conform with experimental observation. This ad hoc series of compromises with classical theory---now referred to as 'the Old Quantum Theory'---left a lot to be desired, and there was a growing call for a systematic overhaul of the kinematic and dynamic formulation of quantized systems. In his groundbreaking 1925 paper "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen", Werner Heisenberg provided a new formalism that would later be refined into Matrix Mechanics. In this short talk, I will outline the results of Heisenberg's seminal paper; and, for context, compare it to the Old Quantum Theory.
Paco Rilloraza: "An FMM-accelerated, high-order boundary integral solver for Taylor states in stellarators"
Stellarators are fusion reactors that use nonaxisymmetric magnetic fields to confine fusion plasma in a roughly toroidal geometry. When designing a stellarator, one must ensure that the magnetic field satisfies magnetohydrodynamic (MHD) equilibrium. In one strategy for solving for MHD equilibrium in a given domain, the domain is divided into nested toroidal regions, and in each region, the magnetic field must be a Beltrami vector field; when the magnetic field satisfies this condition, the plasma is said to be in a Taylor state. One code that utilizes this solution strategy is BIEST (Boundary Integral Equation Solver for Taylor States), written by Malhotra et. al. In this scheme, the boundary-value problem defining the magnetic field of a Taylor state is recast as a boundary integral equation. Building on that scheme, we present a new code that accelerates the integral equation solution using the fast multipole method (FMM) and a fast surface PDE solver. This code is currently a work in progress, but we provide evidence of high-order convergence with some preliminary numerical examples.