Author ORCID Identifier

https://orcid.org/0000-0002-3419-892X

Semester

Fall

Date of Graduation

2024

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Physics and Astronomy

Committee Chair

Zachariah Etienne

Committee Co-Chair

Maura McLaughlin

Committee Member

Maura McLaughlin

Committee Member

Miguel Bezares

Committee Member

Sean McWilliams

Abstract

Numerical relativity (NR) simulations are essential for self-consistently modeling compact binary mergers, whose observations form the heart of gravitational-wave science. Stable NR simulations are the result of mathematical and computational breakthroughs spanning several decades: early attempts to numerically solve the two-body problem in General Relativity (GR) date back to the 1960s, while the first successful black-hole binary merger simulations were performed in 2005 with the generalized harmonic coordinates, and in 2006 using the moving-puncture gauge.

Today, NR continues to drive progress in gravity research, with ongoing efforts focused on increasing physical realism in simulations containing matter, exploring alternative theories of gravity, optimizing numerical algorithms, and fully leveraging the power of modern computer hardware.

Stable numerical evolutions require splitting the governing partial differential equations (PDEs) into two distinct groups: constraint equations, the elliptic PDEs that are solved to generate the initial data, and evolution equations, the hyperbolic PDEs that are integrated forward in time to simulate dynamical spacetimes. Solving elliptic and hyperbolic PDEs requires different numerical techniques, and specialized NR codes have been developed to handle each type of equation. This typically results in an overspecialization in the field of NR, with individual researchers focusing on a single kind of PDE solver.

In this thesis, I introduce NRPyElliptic, a new hyperbolic relaxation solver for elliptic equations developed within the NRPy framework. The hyperbolic relaxation method transforms elliptic PDEs into hyperbolic PDEs, which are then evolved using a combination of damping and wave propagation. The system is evolved until it reaches a steady state, at which point the solution to the hyperbolic PDEs coincides with that of the original elliptic equations. By using standard wave-equation techniques to solve the constraint equations, we can leverage existing expertise and code infrastructure from evolution solvers, allowing development efforts to focus on new physical applications.

NRPyElliptic has been applied to solve different elliptic systems. In its first application, we have generated binary black-hole (BBH) initial data using the conformal flatness approximation. The solution was validated against TwoPunctures, a widely used NR initial data solver, showing excellent agreement. Since conformally flat initial data can only model BBHs with dimensionless spin parameters up to chi=0.93, we drop this assumption in the second application and implement the full coupled elliptic system of Hamiltonian and momentum constraints in conformally curved spacetimes. Extensive validation and convergence tests for single black-hole initial data in conformally curved spacetimes are presented, along with a proof-of-principle solution for the binary case. As high-resolution conformally curved BBH initial data requires a multi-patch numerical domain, the code has been designed with the ability to integrate into such a framework. Finally, as an example of its flexibility, NRPyElliptic is extended to generate static binary solutions for scalar field configurations in k-essence theory.

Download

Share

COinS