Numerical methods for collapsing gravitational waves

Diese Arbeit befasst sich mit der numerischen Evolution von kollabierenden, achsensymmetrischen Gravitationswellen. Von besonderem Interesse ist hierbei die Schwelle zur Entstehung eines Schwarzen Lochs, an welcher kritische Phänomene erwartet werden. Im ersten Teil der Arbeit wird der BAM Code verwendet um Brill-Wellen zu evolvieren. Dieser implementiert die BSSN Gleichungen in Kombination von Moving-Puncture Koordinaten und Finiten-Differenzen. Bei Simulationen mit diesem Setup scheinen Koordinaten-Singularitäten zu entstehen. Es ist daher nicht geeignet, um den Kollaps von Gravitationswellen zu untersuchen. Geometrische Überlegungen zeigen, dass die grundlegende Struktur von Brill-Wellen-Anfangsdaten vom Vorzeichen ihres Amplituden-Parameters abhängt. Wählt man den Amplituden-Parameter negativ, so kann der Kollaps der Daten zu einem Schwarzen Loch erfolgreich simuliert werden. Nahe des kritischen Bereichs schlagen aber auch diese Simulationen fehl.Im zweiten Teil der Arbeit wird der Bamps-Code präsentiert. Dieser implementiert die GHG-Gleichungen mit Hilfe einer pseudospektralen Methode auf einer 3d-Domäne. Anhand einer Vielzahl von Experimenten werden verschiedene Aspekte des Codes, wie dessen Konvergenz, Rechenleistung und die implementierten Randbedingungen, untersucht.Die Simulationen von Brill-Wellen mit Bamps erlauben einen Vergleich mit publizierten Arbeiten. Fast alle Ergebnisse können reproduziert werden. Einzig die Resultate von Sorkin aus dem Jahr 2010 sind im Widerspruch.Der kritische Bereich von Brill-Wellen konnte auf den Amplitudenbereich zwischen 4.696 und 4.698 eingegrenzt werden. Im Rahmen dieser Genauigkeit war es noch nicht möglich, kritisches Verhalten zu beobachten. Im superkritischen Bereich wurde festgestellt, dass nahe der kritischen Amplitude zwei voneinander getrennte Horizonte entstehen, die resultierende Raumzeit also zwei Schwarze Löcher zu enthalten scheint

Constraint damping for the Z4c formulation of general relativity

One possibility for avoiding constraint violation in numerical relativity simulations adopting free-evolution schemes is to modify the continuum evolution equations so that constraint violations are damped away. Gundlach et. al. demonstrated that such a scheme damps low amplitude, high frequency constraint violating modes exponentially for the Z4 formulation of General Relativity. Here we analyze the effect of the damping scheme in numerical applications on a conformal decomposition of Z4. After reproducing the theoretically predicted damping rates of constraint violations in the linear regime, we explore numerical solutions not covered by the theoretical analysis. In particular we examine the effect of the damping scheme on low-frequency and on high-amplitude perturbations of flat spacetime as well and on the long-term dynamics of puncture and compact star initial data in the context of spherical symmetry. We find that the damping scheme is effective provided that the constraint violation is resolved on the numerical grid. On grid noise the combination of artificial dissipation and damping helps to suppress constraint violations. We find that care must be taken in choosing the damping parameter in simulations of puncture black holes. Otherwise the damping scheme can cause undesirable growth of the constraints, and even qualitatively incorrect evolutions. In the numerical evolution of a compact static star we find that the choice of the damping parameter is even more delicate, but may lead to a small decrease of constraint violation. For a large range of values it results in unphysical behavior.Comment: 13 pages, 24 figure

Solving 3D relativistic hydrodynamical problems with WENO discontinuous Galerkin methods

Discontinuous Galerkin (DG) methods coupled to WENO algorithms allow high order convergence for smooth problems and for the simulation of discontinuities and shocks. In this work, we investigate WENO-DG algorithms in the context of numerical general relativity, in particular for general relativistic hydrodynamics. We implement the standard WENO method at different orders, a compact (simple) WENO scheme, as well as an alternative subcell evolution algorithm. To evaluate the performance of the different numerical schemes, we study non-relativistic, special relativistic, and general relativistic testbeds. We present the first three-dimensional simulations of general relativistic hydrodynamics, albeit for a fixed spacetime background, within the framework of WENO-DG methods. The most important testbed is a single TOV-star in three dimensions, showing that long term stable simulations of single isolated neutron stars can be obtained with WENO-DG methods.Comment: 21 pages, 10 figure

Collapse of Nonlinear Gravitational Waves in Moving-Puncture Coordinates

We study numerical evolutions of nonlinear gravitational waves in moving-puncture coordinates. We adopt two different types of initial data -- Brill and Teukolsky waves -- and evolve them with two independent codes producing consistent results. We find that Brill data fail to produce long-term evolutions for common choices of coordinates and parameters, unless the initial amplitude is small, while Teukolsky wave initial data lead to stable evolutions, at least for amplitudes sufficiently far from criticality. The critical amplitude separates initial data whose evolutions leave behind flat space from those that lead to a black hole. For the latter we follow the interaction of the wave, the formation of a horizon, and the settling down into a time-independent trumpet geometry. We explore the differences between Brill and Teukolsky data and show that for less common choices of the parameters -- in particular negative amplitudes -- Brill data can be evolved with moving-puncture coordinates, and behave similarly to Teukolsky waves

Solving 3D relativistic hydrodynamical problems with weighted essentially nonoscillatory discontinuous Galerkin methods

Discontinuous Galerkin (DG) methods coupled to weighted essentially nonoscillatory (WENO) algorithms allow high order convergence for smooth problems and for the simulation of discontinuities and shocks. In this work, we investigate WENO-DG algorithms in the context of numerical general relativity, in particular for general relativistic hydrodynamics. We implement the standard WENO method at different orders, a compact (simple) WENO scheme, as well as an alternative subcell evolution algorithm. To evaluate the performance of the different numerical schemes, we study nonrelativistic, special relativistic, and general relativistic test beds. We present the first three-dimensional simulations of general relativistic hydrodynamics, albeit for a fixed spacetime background, within the framework of WENO-DG methods. The most important test bed is a single Tolman-Oppenheimer-Volkoff star in three dimensions, showing that long term stable simulations of single isolated neutron stars can be obtained with WENO-DG methods