Non-linear numerical Schemes in General Relativity

TL;DR

This thesis develops advanced numerical methods to solve Einstein's equations, demonstrating stable simulations of cosmic strings, gravitational waves, and neutron star oscillations with high accuracy, advancing computational relativity.

Contribution

It introduces a stable second order convergent Cauchy characteristic matching code, fully nonlinear evolutions of cosmic strings, and a high-precision scheme for neutron star oscillations, enhancing numerical relativity tools.

Findings

Stable long-term evolution of gravitational fields in cylindrical symmetry.

First fully nonlinear simulations of cosmic strings in curved spacetime.

High-accuracy modeling of neutron star radial oscillations and mode coupling.

Abstract

This thesis describes the application of numerical techniques to solve Einstein's field equations in three distinct cases. First we present the first long-term stable second order convergent Cauchy characteristic matching code in cylindrical symmetry including both gravitational degrees of freedom. Compared with previous work we achieve a substantial simplification of the evolution equations as well as the relations at the interface by factoring out the z-Killing direction via the Geroch decomposition in both the Cauchy and the characteristic region. In the second part we numerically solve the equations for static and dynamic cosmic strings of infinite length coupled to gravity and provide the first fully non-linear evolutions of cosmic strings in curved spacetimes. The inclusion of null infinity as part of the numerical grid allows us to apply suitable boundary conditions on the metric and the matter fields to suppress unphysical divergent solutions. The code is used to study the interaction between a Weber-Wheeler pulse of gravitational radiation with an initially static string. In the final part of the thesis we probe a new numerical approach for highly accurate evolutions of non-linear neutron star oscillations in the case of radial oscillations of spherically symmetric stars. For this purpose we view the evolution of the physical quantities as deviations from a static equilibrium configuration and reformulate the equations in a fully non-linear perturbative form. The high accuracy of the new scheme enables us to study the non-linear coupling of eigenmodes over a wide range of initial amplitudes.