Evolving Brill wave spacetimes using Carpet
Peter Diener, Luis Lehner, Dave Neilsen, Manuel Tiglio and Jason Ventrella
Brill wave spacetimes have been shown in 2D to exhibit critical behaviour. The resolution requirements to show this are so severe that it has not been possible to reproduce these results in 3D. In this talk I will present some recent results on the evolution of axi-symmetric brill wave spacetimes using 3D codes with Fixed Mesh Refinement (FMR). In addition to FMR the cartoon technique is used to further extend the range of achievable resolutions.
Dynamical control of constraint growth and gauges in numerical evolutions of Einstein's equations
Ernst Nils Dorband
A popular method to gain better stability in the numerical evolution of space times is to construct parametrized adjustments to the evolution equations with suitable parameters that improve the system. An example would be to add multiples of the constraints to the equations. Recently there have been developments that allow to control the growth of some global norm of the constraints by dynamically choosing the parameters during the run in a way that the constraint norm follows a chosen behaviour. We show results of experiments with such a method implemented in the BSSN system. In the same spirit we look at the possibility to find gauge conditions that could for example minimize certain gradients of field variables in a dynamical way.
Shift Conditions for Orbiting Black Holes in Numerical Relativity
F Siddhartha Guzmán
We present a simple scheme to create a co-rotating coordinate system for orbiting binary systems in numerical relativity. The scheme is able to find a shift vector that brings orbiting binary systems into a co-rotating frame with respect to the coordinate system used to do the computations. Furthermore, it is able to track the orbiting system, adjusting automatically to account for changes in orbital velocity, as will be encountered during binary inspirals. We demonstrate how this scheme works in a binary black hole evolution in full numerical relativity and explore the conditions under which it is optimal for this particular case. The scheme is fairly general, and should be applicable to binary systems of many kinds, including binary black holes and neutron stars.
Extracting Gravitational Radiation from Numerical Evolutions
Frank Hermann
Different techniques for radiation extraction from numerical simulations of 3D black hole collisions are discussed. The Newman-Penrose scalar Psi_4 and the Zerilli quantity numerically extracted from a head-on collison are compared and the usefulness of other (gauge dependent) quantities is mentioned.
Numerical studies of the merger of black hole binaries from close quasi-circular orbits
Michael Koppitz, Denis Pollney, Erik Schnetter, Ed Seidel
The late stages of binary black hole inspiral involves a transition from a slow inspiral, accurately modelled by post-newtonian calculations, to a plunge where the full relativistic theory must be taken into account. A number of proposals exist for solving the momentum constraints on an initial hypersurface for the case of binary black holes in the late stages of inspiral, ie. in quasi-circular orbit but at close separations. One approach is to solve the thin-sandwich equations under the additional assumption of a helical killing vector (HKV) to model the quasi-stationarity. This approach has been applied by Grandclement et al. (2002) to produce initial data sets in good agreement with post-newtonian predictions of the innermost quasi-circular orbit.
This talk presents the results of first simulations of equal-mass binary black hole mergers from initial data produced via the thin-sandwich HKV technique. The runs are followed past merger, as indicated by the appearance of a common apparent horizon. We describe gauge conditions appropriate for these models, in particular a dynamically adjusted co-rotating frame allows the individual apparent horizons to be held essentially in place, postponing the need to handle moving excision regions. Accurate determination of the dynamics is strongly dependent on grid resolution. The co-rotating frame allows for regions of higher resolution ("fixed mesh refinement") in the neighbourhood of the horizons. The results of these simulations from thin sandwich data are compared with recent simulations of puncture black holes. We study various aspects of the horizon dynamics, using tools such as the isolated horizon formalism, to determine measures of black hole mass and spin, as well as discuss prospects for wave extraction.
Excision boundary conditions for relativistic hydrodynamics
Frank Loeffler
Numerical simulations of neutron star collapse and black hole - neutron star binary systems are significantly complicated by the presence of matter fields near the singularity. It is often very difficult to evolve the region near the singularity in a numerical simulation. In evolutions of vacuum spacetimes {\it excision} techniques are used to remove the parts of the spacetime that are causally disconnected from the domain of interest. Excision of a vacuum spacetime is usually done by extrapolating from the known to the unknown region. This method does not generalize robustly to the matter fields.
In order to evolve a spacetime containing fluid matter conservative shock capturing methods are usually required. These methods ensure the correct solution is found near shocks. However, the requirements of these methods mean that the implementation of excision is not as straightforward as in the vacuum case.
We present a method for applying excision boundary conditions that is consistent with the conservative nature of the hydrodynamics. This technique is applied to a variety of shock capturing schemes, producing stable and convergent evolutions in static and dynamic situations even in the presence of a shock. We also present results showing how these techniques can be used to extend the lifetime of simulations of neutron star collapse to significantly past black hole formation.
Recent Progress in General-Relativistic Simulations of Rotational Stellar Core Collapse
Christian David Ott
The collapse of the iron core of a rotating core-collapse supernova progenitor star is one of the most promising sources of gravitational waves. Recently, much progress towards reliable theoretical estimates of the gravitational wave signature of such an event has been made. This talk will summarize these advances and present in turn new results from a set of fully general-relativistic three-dimensional simulations of rotational core collapse that we have performed. Our calculations make use of the Cactus Toolkit in combination with mesh refinement. The spacetime is evolved with the BSSN evolution system and general relativistic ideal fluid hydrodynamics are implemented using high-resolution shock capturing methods.
Analysing Horizons in Numerical Relativity
Erik Schnetter
We present methods to analyse the world tubes of apparent horizons in numerical simulations. As our intent is to perform this analysis locally in time instead of via postprocessing, we cannot know the location of the event horizon. Remaining local in time allows us to obtain results while a simulation is performed, making it possible to feed back into the simulation and influence gauge or boundary conditions. This also allows us to analyse simulations that are not stable enough to reach a stationary final state.
We use the Isolated and Dynamical Horizon formalism introduced by Ashtekar et al. We calculate spin and total mass for horizons with axial Killing vector fields and give an expansion of spin and mass in terms of invariant multipole moments. We also calculate the energy and angular momentum fluxes through the horizon due to matter and gravitational radiation. We present results from 3+1-dimensional simulations for collapsing stellar cores and coalescing black holes.
Black Hole Excision with Multiple Grid Patches
Jonathan Thornburg
When using black hole excision to numerically evolve a black hole spacetime with no continuous symmetries, most $3+1$ finite differencing codes use a Cartesian grid. It's difficult to do excision on such a grid, because the natural $r = \hbox{\rm constant}$ excision surface must be approximated either by a very different shape such as a contained cube, or by an irregular and non-smooth ``LEGO$^{\rm (TM)}$ sphere'' which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme, which uses multiple $\{ r \times (\hbox{\rm angular coordinates}) \}$ grid patches, each patch using a different (nonsingular) choice of angular coordinates. This allows excision on a smooth $r = \hbox{\rm constant}$ 2-sphere.
I discuss the key design choices in such a multiple-patch scheme, including the number and shape of the patches (I use a 6-patch ``inflated-cube'' scheme), the way in which the patch ghost zones are ``synchronized'' by interpolation from neighboring patches, the tensor basis for the Einstein equations in each patch, and the handling of non-tensor field variables such as the BSSN $\tilde{\Gamma}^i$ (I use a scheme which requires ghost zones which are twice as wide for the BSSN conformal factor $\phi$ as for $\tilde{\Gamma}^i$ and the other BSSN field variables).
Finally, I present sample numerical results from a prototype implementation of this scheme. This code simulates the time evolution of the (asymptotically flat) spacetime around a single (excised) black hole. Using Kerr initial data with $J/m^2 = 0.6$, I present evolutions to beyond $t = 1000m$. The lifetime of these evolutions appears to be limited only by instabilities at the outer boundary (where I have only partially implemented outgoing-radiation boundary conditions), and {\em not\/} by any instabilities due to the multiple-patch scheme.