Session R8 - Focus Session: Numerical Relativity I.
FOCUS session, Monday afternoon, April 07
Regency Ballroom C2, Loews Philadelphia Hotel

[R8.001] Numerical Simulation of Black Holes

Einstein's equations of general relativity are prime candidates for numerical solution on supercomputers. There is some urgency in being able to carry out such simulations: Large-scale gravitational wave detectors are now coming on line, and the most important expected signals cannot be predicted except numerically.

Problems involving black holes are perhaps the most interesting, yet also particularly challenging computationally. One difficulty is that inside a black hole there is a physical singularity that cannot be part of the computational domain. A second difficulty is the disparity in length scales between the size of the black hole and the wavelength of the gravitational radiation emitted. A third difficulty is that all existing methods of evolving black holes in three spatial dimensions are plagued by instabilities that prohibit long-term evolution.

I will describe the ideas that are being introduced in numerical relativity to deal with these problems, and discuss the results of recent calculations of black hole collisions.

[R8.002] The final plunge of spinning binary black holes

With the goal of bringing theory, particularly numerical relativity, to bear on an astrophysical problem of critical interest to gravitational wave observers we introduce a model for coalescence radiation from binary black hole systems. We build our model using the Lazarus approach, a technique that bridges far and close limit approaches with full numerical relativity to solve Einstein equations applied in the truly nonlinear dynamical regime. We specifically study the post-orbital radiation from a system of equal-mass non-spinning black holes, deriving waveforms which indicate strongly circularly polarized radiation of roughly 3% of the system's total energy and 12% of its total angular momentum in just a few cycles. Supporting this result we first establish the reliability of the late-time part of our model, including the numerical relativity and close-limit components, with a thorough study of waveforms from a sequence of black hole configurations varying from previously treated head-on collisions to representative target for ``ISCO'' data corresponding to the end of the inspiral period. We then complete our model with a simple treatment for the early part of the spacetime based on a standard family of initial data for binary black holes in circular orbit. A detailed analysis shows strong robustness in the results as the initial separation of the black holes is increased from 5.0 to 7.8M supporting our waveforms as a suitable basic description of the astrophysical radiation from this system. Finally, a simple fitting of the plunge waveforms is introduced as a first attempt to facilitate the task of analyzing data from gravitational wave detectors.

[R8.003] Evolving a Black Hole with Mesh Refinement

We carry out evolutions of a single black hole in 3-D with mesh refinement. We evolve the black hole as a "puncture" using a BSSN system with various slicing conditions. We explore the behavior of the solution near the mesh refinement boundary, and the evolution of the puncture at high resolution.

[R8.004] Dynamic singularity excision

In the context of the ongoing goal of simulating a binary black-hole evolution through merger and beyond, we discuss algorithmic developments of our dynamic singularity excision techniques. By allowing the holes to move across the computational domain (rather than keeping them pinned to the grid and treating their motion via gauge choices), we must devise methods for providing data for points which had previously been excised, i.e. ``populating'' these points with valid data. We report on new developments in this work, with emphasis on maintaining the smooth nature of the solutions throughout re-population.

[R8.005] Head-On Binary Black-Hole Collisions in BSSN

We re-visit the binary black-hole head-on collision using the Penn State "Maya" code. Evolutions are in full 3D, using the BSSN system of evolution equations. Singularities are handled with dynamic excision; their locations can be determined from the dynamical variables by multiple means, and fed back to update the location of the excision region. It is expected that this will be a useful test of the ability of Maya to deal with multiple black holes in full 3D.

[R8.006] Constraint violations in 3D black hole evolutions and boundary conditions

Numerical evolutions of black holes and other spacetimes typically suffer from constraint violating modes that grow without bound. Here we numerically study the influence on these modes of the fact that standard boundary conditions in numerical relativity are inconsistent with the constraints. More specifically, we numerically study the evolution of the constraints itself, as a set of symmetric hyperbolic equations, when linearized around a 3D Schwarzschild black hole. The goal of these studies is to determine whether constraint-preserving boundary conditions would improve the stability of unconstrained numerical evolutions.

[R8.007] Analysis of mode-mode coupling in black hole waveforms

Using a fully nonlinear characteristic treatment, we compute gravitational waveforms from single-black hole spacetimes. In this approach, initial data is given on an outgoing null cone approximating past null infinity, with boundary data prescribed on the past horizon. We perform code calibration tests and carry out a comparison with a perturbative characteristic treatment. By performing a decomposition of the fully nonlinear gravitational radiation signal in terms of spin-weighted spherical harmonics, we can make quantitative statements about the degree of nonlinearity and the effect of mode-mode coupling. We present sample results and an outline of future work.

[R8.008] Treating star-black hole systems via the characteristic formulation of GR

We discuss two projects towards treating black hole-star systems numerically via the characteristic approach. Both supermassive black holes and solar mass black holes can be treated.

[R8.009] Quasiequilibrium models for binary black hole systems

A numerical scheme for constructing quasiequilibrium orbits for binary black hole systems is described. We use a second order finite difference method for the solution of the elliptic equations associated with the Hamiltonian and momentum constraints. The initial data are computed using a superposition of two Kerr-Schild black holes. We performed several convergence tests and we compared the results of this new code with other existing computations.

[R8.010] Binary systems with helical Killing vectors

We consider compact binary systems, modeled as spacetimes with a helical Killing vector, heuristically the generator of time-translations in a corotating frame. Systems that are stationary in this sense have equal amounts of ingoing and outgoing radiation. Work is described on the numerical construction of helically symmetric binary black hole and binary neutron star systems. Analytic models of point particles bound by scalar, electromagnetic, and post-Minkowskian gravitational fields are used to estimate the error in using the approximation of helical symmetry.

[R8.011] Towards a Realistic Neutron Star Binary Inspiral: Initial Data and Long Timescale Evolution

I present results produced by my fully consistent general relativistic hydrodynamics numerical relativity code. First, I analyze the conformally flat, quasiequilibrium (CFQE) sequence approximation by directly comparing it to solutions of the Einstein equations obtained numerically using CFQE configurations as initial data. In this way, I demonstrate how one can go about constructing astrophysically relevant initial data sets. I then perform the first detailed analysis of the characteristics of the orbital motion of finite sized compact objects in numerical relativity in full 3+1 general relativistic calculations.

[R8.012] Dynamical Determination of the ISCO for Neutron Star Binaries

We report on the latest results emerging from numerical simulations of neutron-star binary systems in circular orbit generated with our new, fully relativistic hydrodynamical code in 3+1. The initial data sets are conformally flat, quasi-equilibrium solutions to the Hamiltonian and momentum constraints obtained by adopting the Wilson-Mathews scheme (i.e. the ``thin-sandwich'' approach). The solutions correspond to equal mass, corotating and irrotational binaries in circular orbit constructed from a polytropic equation state. The initial models are evolved in a corotating frame using the BSSN implementation of the field equations under a variety of gauge choices. The hydrodynamical variables are updated through a general relativistic version of the Van Leer method. We perform several simulations of the same binary stars orbiting at different separations to determine dynamically the innermost stable circular orbit (ISCO). Typically, stable orbits are found to remain in near-circular equilbrium for over two orbital periods, at which point we terminate the integrations, while unstable orbits plunge inward rapidly in less than a period.

[R8.013] Toward standard testbeds for numerical relativity

Many different numerical evolution schemes for Einstein's equations have been proposed to address stability and accuracy problems. Differences in performance originate from many sources, including formulations of the equations, gauges, boundary conditions, numerical methods, and so on. Approaches that seem effective for one set of initial data often fail when applied to another. Based on the collective experience of researchers from a dozen groups worldwide, we propose a test suite for comparing numerical evolution codes that is designed to probe their strengths and weaknesses and to separate out the different effects, and their causes, seen in the results. In some cases, the testbed may give insight to help explain the results but, even lacking that, it provides the basis for clear phenomenological comparisons. The test suite is presently limited to vacuum spacetimes, can be run on modest computational resources, and can be used with many different approaches. The tests, and their output, are completely and precisely specified so that they may be used to make clear and meaningful comparisons.

[R8.014] Evolving excised black holes with TVD numerical methods

Total Variation Diminishing (TVD) numerical methods have improved stability properties for nonlinear differential equations, and are widely used in computational fluid dynamics. While Einstein's equations are not genuinely nonlinear, these methods may be advantageous for solving the Einstein equations in specific instances, such as evolving fluid spacetimes and black holes with excision. Using a Frittelli-Reula formulation of the Einstein equations, I will present results of 1-D and 3-D black hole evolutions, and compare the performance of TVD methods with other numerical approaches.

[R8.015] Dirty Black Hole Evolutions

Reaching the ultimate goal of simulating binary black hole systems, with enough accuracy and dynamical range to be relevant to observations, requires careful development and implementation of infrastructure to handle all aspects of the problem. Gauge conditions, initial data, boundary conditions, black hole singularities, evolution equations,constraint preservation and numerical approximations are a few examples. At the same time, insight also can be obtained by following a "quick and dirty" approach in which some elements of the problem are simplified or approximated. We present results of binary black hole evolutions following this approach. The quality of these results are limited because of their quick and dirty nature, but they provide valuable information for future, cleaner calculations.

[R8.016] Rigorous numerical techniques for stable 3D black hole evolutions

The discretization of partial differential equations near boundaries, in such a way that numerical stability is achieved, is a challenging problem that appears frequently in numerical relativity. The presence of corners, edges and excision boundaries adds even further complications. Here we make use of rigorous techniques from numerical analysis to construct discrete derivative and dissipative operators for 3D evolutions in the presence of black holes that guarantee stability of the semidiscrete problem. Rigorous techniques for imposing boundary conditions at the discrete level, even for non-smooth domains, are also introduced. As a testbed, 3D evolutions of a scalar field and Maxwell equations around a Schwarzschild black hole are presented.

Part R of program listing