Date of Award

2012

Degree Type

Open Access Dissertation

Degree Name

Computational Science Joint PhD with San Diego State University, PhD

Program

School of Mathematical Sciences

Advisor/Supervisor/Committee Chair

Ricardo Carretero-González

Dissertation or Thesis Committee Member

Peter Blomgren

Dissertation or Thesis Committee Member

Michael Bromley

Dissertation or Thesis Committee Member

Ali Nadim

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Rights Information

© 2012 Ronald Meyer Caplan

Abstract

We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics.

Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size.

The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.

Comments

Fourth Committee Member: Allon Percus

DOI

10.5642/cguetd/52

 
 
 
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