Welcome
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Meetings
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Methods
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Results
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Schedule
The
main topic of the workshop is code comparison in
numerical relativity but we welcome anyone with analytic expertise in
any of the underlying issues affecting numerical stability: hyperbolic
formulations of Einstein's equations,
boundary conditions, constraint preservation, gauge conditions. Talks
of such general interest will be scheduled on the first three or four
days of the workshop. Contact Manuel Tiglio
(tiglio_at_astro.cornell.edu), Oscar Reula (reula_at_fis.uncor.edu) or
Jeff Winicour (jeff_at_einstein.phyast.pitt.edu) to be put on the
schedule.
The
talks announced so far for this section are the following (in
alphabetical order).
- Tilman Vogel, Tuesday 11:00A
Title: Stability and
constraint propagation
Authors: Tilman Vogel,
AEI, Golm
and: Joerg Frauendiener,
Institut
für Astronomie und Astrophysik, Universität Tübingen
Abstract: A typical problem of numerical relativity is the instable
behaviour of simulations. Growing constraint quantities destroy the
physical properties of the solutions on short time-scales.
Two main sources are connected to the formulation of the IBVP: (i)
Inconsistent boundary conditions inject unphysical data into the
numerical domain. (ii) Small but unavoidable constraint violations in
the initial data trigger instabilities of the evolution equations. The
latter problem, which is already part of the continuum formulation,
will
be discussed in this talk.
It will be sketched, how a stability analysis for such perturbations
can
be carried out. The essential influence of the choice of the foliation
will be demonstrated. As examples, analytical results for the Maxwell
equations and the Bianchi equation will be presented.
[1] Joerg Frauendiener and Tilman Vogel: On the stability of constraint
propagation, gr-qc/0410100
- Bernd
Reimann. Tuesday 11:00AM
Instituto de Ciencias Nucleares, Universidad Nacional
Aut{\'o}noma de M{\'e}xico, A.P. 70-543, M{\'e}xico D.F. 04510,
M{\'e}xico
Max Planck Institut f\"ur Gravitationsphysik,
Albert Einstein Institut, Am M\"uhlenberg 1, 14476 Golm, Germany
Title: Avoiding
Gauge and Constraint Shocks in 1D Numerical Relativity
Abstract:
I will report on recent work (gr-qc/0411094) done together with M.
Alcubierre,
J.A. Gonzalez and D. Nunez:
For evolution systems of the type found in Numerical Relativity we
successfully
make use of mathematical theory regarding the singularity formation in
wavelike
hyperbolic equations. This theory states that for systems of partial
differential equations two kinds of shocks can be expected to form,
namely
"gradient catastrophes" and "blow-ups within finite time". For both
mechanisms
we are able to derive shock-avoiding conditions which we refer to as
"indirect
linear degeneracy" and "source criteria". For a toy problem, the wave
equation
with dynamic wave speed and quadratic source terms, we show that the
solution
has wavelike and shock-avoiding character only if both conditions are
satisfied.
In our primary example, Einstein's equations in spherical symmetry, the
two
criteria detect "gauge and constraint shocks". Here the gauge shocks are
associated with the foliation itself, and the shock formation in the
evolution
variables can be avoided by choosing the lapse and the shift function
appropriately. The constraint shocks arise due to numerically
unfavorable terms
in the evolution equations and cause constraint violations to become
infinite
within finite time. For the ADM system in spherical symmetry, we show
that
these shocks can be avoided by adding constraints to some evolution
equations.
By doing so, we obtain a 1-parameter family of shock-avoiding evolution
systems
which in numerical experiments such as the robust stability test shows
excellent
numerical behavior.
- Osvaldo
Moreschi, Tuesday 3PM
FaMAF, Universidad de Cordoba,
Argentina.
Ttitle: An interior-asymptotics
approach to the binary black hole problem
author: Osvaldo M. Moreschi
abstract:
We present an approach to the
binary black hole problem that makes
use of two complementary
techniques.
We study the dynamical equations
for a binary system of compact
objects arising from the
asymptotic structure of the corresponding
isolated system. We make use of
the center of mass frame at
future null infinity and keep
terms up to first order of
gravitational radiation.
We complement this system with
the equations of motion of two
black holes, in the interior of
the spacetime.
- Marronetti,
Pedro, Wednesday 9:30AM
Florida Atlantic University
Title: "Binary Neutron Star
Simulations as Testing Ground for
Numerical Relativity".
Abstract: New formulations
for the evolution of gravitational fields in numerical
relativity are continuously
presented to the scientific community. Their
main goal is to achieve stable
and reliable evolutions of compact-object
binaries. However, due to the
complexity of the required numerical work,
few of the many formulations
found in the literature have been tested on
binary evolutions. We introduce
a new testing ground for numerical
methods based on the simulation
of binary neutron stars (BNS). Our
objective is to benchmark new
formalisms against the currently most
stable simulations. BNS
simulations usually require extensive
computational resources and the
length of the runs could, in principle,
render these tests impractical.
We show how small, low resolution grids
can be used to gain insight into
the stability of different numerical
schemes, with runs that only
take a few hours on single-processor
workstations.
- Carles
Bona, Wednesday 11AM
UIB, Palma-Spain
Title: "Covariant freezing shift
conditions"
The well known 'gamma-freezing' shift condition is generalized in a
covariant way. Two lapse conditions are also considered in this
context. The relationship of these gauge conditions with the existence
of a time Killing vector is pointed out. Two simple test cases are
investigated numerically: the 1D Kerr-Schild Black Hole and the
Minkowsky gauge waves.
- Sascha
Husa, Wednesday 3PM
AEI
TBA
- Santiago
Gomez, Wednesday 4PM
FaMAF, Cordoba.
Title: Dynamics of tetrad fields of Schwarzchild spacetime
In
tetrad formalism, the Einstein equations can be reduced to a simple
system of
ordinary differential equations. The system is made up of 30
scalar
fields, in addition there are 30 constrains to be satisfied by the
system. The
evolution of the constrains have the same stability property
as the
original fields. The gauge data is provided by the temporal
projection
of the four tetrad vectors and six connections vectors. The
system have
a natural unstable behavior unless a trivial and special gauge
is taken,
when the temporal projection of the connection are assumed to be
null. Only
for static frames we found a way of having a system with stable
behavior
without that special and trivial gauge, it consists of changing
the sign of
the positive eigenvalues of the system. Into the evolution
equation of
the fields appears some components of the Weyl tensor with the
last two
indices contracted with the tetrad, they are fixed by calculing
them for the
case of Schwarzschild spacetime. The initial data for all
fields are
obteined in the same way. The angular part of the tetrad will
be separated
into six diferent patches using cubito coordinate. Due the
spherical
symmetry of the system, all patches will have the same
behavior.
In a near
future we expect to be ready to evolve non-spherical symmetric
spacetimes
into this formalism.
- Jeff
Winicour, Wednesday 4:30PM
Pittsburgh University and AEI
Shifted Gauge Wave Simulations
There will also be presentations by representatives of the groups
carrying out code comparison. These talks will explain the formalism
underlying the code, its numerical implementation, and its results on
the standardized tests. At
present, the following people have committed to presenting such talks:
- Carles Bona
- Sascha Husa
- Carlos Palenzuela
- Denis Pollney
- Bela Szilagyi
- Manuel Tiglio
- Jeff Winicour
The second week of the workshop will be devoted to carrying out code
tests and comparisons and to the design of further tests. The code
comparisons will focus on:
- The
first round of Mexico tests with periodic boundary conditions, with
the proposed modifications proposed to run at higher amplitudes
and with
- Runge-Kutta
integrators as replacement for ICN.
- Mexico
tests for a gauge wave with shift.
- The
first round of boundary tests.
- The
design of new tests will focus on black hole spacetimes.
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