Numerical Relativity and Inhomogeneous Cosmologies

Abstract

In this work numerical methods for solving Einstein's equations are developed and applied to the study of inhomogeneous cosmological models. A two-dimensional computer code is described which implements two advanced numerical methods: LeVeque's multi-dimensional high-resolution integration scheme which allows accurate evolution of solutions containing discontinuities or steep gradients, and an adaptive mesh refinement (AMR) algorithm which enables the local resolution of a simulation to vary dynamically in response to the behaviour of the evolved solution. A family of hyperbolic formulations of the Einstein equations is derived by generalization of an evolution system proposed by Frittelli and Reula, and numerical solutions produced using these formulations are compared to solutions produced using alternative reductions of the evolutions equations. Properties of the harmonic time slicing condition are also investigated, and analytic and numerical results concerning the formation of coordinate singularities are presented. Numerical simulations are performed of planar cosmologies, described using Gowdy's reduction of the Einstein equations, and U(1)-symmetric cosmologies, described using Moncrief's reduction of the Einstein equations, with the spacetimes in both cases being vacuum. Numerical studies follow up on the work of Berger, Moncrief and co-workers, with attention being focused on the small-scale features that develop in the models and the behaviour of linear and nonlinear gravitational waves.

PhD dissertation. TeXsis, 203 pages, 39 figures including 6 colour figures (pages 26,53,128,129,140,181) and a colour version of one figure (page 54) in JPEG format

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Cited by in corpus (13)

• Numerical Approaches to Spacetime Singularities
• Exploring New Physics Frontiers Through Numerical Relativity
• Oscillatory approach to the singularity in vacuum spacetimes with $T^2$ isometry
• Spike Oscillations
• Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems
• Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators
• New explicit spike solution -- non-local component of the generalized Mixmaster attractor
• Interface Conditions for Wave Propagation Through Mesh Refinement Boundaries
• Axisymmetric Numerical Relativity
• Constraint Preserving Boundary Conditions for Hyperbolic Formulations of Einstein's Equations
• Testing the Cactus code on exact solutions of the Einstein field equations
• Generating spiky solutions of Einstein field equations with the Stephani transformation
• Dual Foliation Formulations of General Relativity