6 research outputs found
Testing numerical relativity with the shifted gauge wave
Computational methods are essential to provide waveforms from coalescing black holes, which are expected to produce strong signals for the gravitational wave observatories being developed. Although partial simulations of the coalescence have been reported, scientifically useful waveforms have so far not been delivered. The goal of the AppleswithApples (AwA) Alliance is to design, coordinate and document standardized code tests for comparing numerical relativity codes. The first round of AwA tests have now being completed and the results are being analyzed. These initial tests are based upon periodic boundary conditions designed to isolate performance of the main evolution code. Here we describe and carry out an additional test with periodic boundary conditions which deals with an essential feature of the black hole excision problem, namely a non-vanishing shift. The test is a shifted version of the existing AwA gauge wave test. We show how a shift introduces an exponentially growing instability which violates the constraints of a standard harmonic formulation of Einstein's equations. We analyze the Cauchy problem in a harmonic gauge and discuss particular options for suppressing instabilities in the gauge wave tests. We implement these techniques in a finite difference evolution algorithm and present test results. Although our application here is limited to a model problem, the techniques should benefit the simulation of black holes using harmonic evolution codes
Initial data transients in binary black hole evolutions
We describe a method for initializing characteristic evolutions of the
Einstein equations using a linearized solution corresponding to purely outgoing
radiation. This allows for a more consistent application of the characteristic
(null cone) techniques for invariantly determining the gravitational radiation
content of numerical simulations. In addition, we are able to identify the {\em
ingoing} radiation contained in the characteristic initial data, as well as in
the initial data of the 3+1 simulation. We find that each component leads to a
small but long lasting (several hundred mass scales) transient in the measured
outgoing gravitational waves.Comment: 18 pages, 4 figure
Simulation of Binary Black Hole Spacetimes with a Harmonic Evolution Scheme
A numerical solution scheme for the Einstein field equations based on
generalized harmonic coordinates is described, focusing on details not provided
before in the literature and that are of particular relevance to the binary
black hole problem. This includes demonstrations of the effectiveness of
constraint damping, and how the time slicing can be controlled through the use
of a source function evolution equation. In addition, some results from an
ongoing study of binary black hole coalescence, where the black holes are
formed via scalar field collapse, are shown. Scalar fields offer a convenient
route to exploring certain aspects of black hole interactions, and one
interesting, though tentative suggestion from this early study is that behavior
reminiscent of "zoom-whirl" orbits in particle trajectories is also present in
the merger of equal mass, non-spinning binaries, with appropriately fine-tuned
initial conditions.Comment: 16 pages, 14 figures; replaced with published versio
The discrete energy method in numerical relativity: Towards long-term stability
The energy method can be used to identify well-posed initial boundary value
problems for quasi-linear, symmetric hyperbolic partial differential equations
with maximally dissipative boundary conditions. A similar analysis of the
discrete system can be used to construct stable finite difference equations for
these problems at the linear level. In this paper we apply these techniques to
some test problems commonly used in numerical relativity and observe that while
we obtain convergent schemes, fast growing modes, or ``artificial
instabilities,'' contaminate the solution. We find that these growing modes can
partially arise from the lack of a Leibnitz rule for discrete derivatives and
discuss ways to limit this spurious growth.Comment: 18 pages, 22 figure
From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
This article reviews some aspects in the current relationship between
mathematical and numerical General Relativity. Focus is placed on the
description of isolated systems, with a particular emphasis on recent
developments in the study of black holes. Ideas concerning asymptotic flatness,
the initial value problem, the constraint equations, evolution formalisms,
geometric inequalities and quasi-local black hole horizons are discussed on the
light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity.
Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24
November, 2006), part of the "General Relativity Trimester" at the Institut
Henri Poincare (Fall 2006). Comments and references added. Typos corrected.
Submitted to Classical and Quantum Gravit
Some mathematical problems in numerical relativity
The main goal of numerical relativity is the long time simulation of highly nonlinear spacetimes that cannot be treated by perturbation theory. This involves analytic, computational and physical issues. At present, the major impasses to achieving global simulations of physical usefulness are of an analytic/computational nature. We present here some examples of how analytic insight can lend useful guidance for the improvement of numerical approaches