435 research outputs found
Testing the Cactus code on exact solutions of the Einstein field equations
The article presents a series of numerical simulations of exact solutions of
the Einstein equations performed using the Cactus code, a complete
3-dimensional machinery for numerical relativity. We describe an application
(``thorn'') for the Cactus code that can be used for evolving a variety of
exact solutions, with and without matter, including solutions used in modern
cosmology for modeling the early stages of the universe. Our main purpose has
been to test the Cactus code on these well-known examples, focusing mainly on
the stability and convergence of the code.Comment: 18 pages, 18 figures, Late
Hyperbolic slicings of spacetime: singularity avoidance and gauge shocks
I study the Bona-Masso family of hyperbolic slicing conditions, considering
in particular its properties when approaching two different types of
singularities: focusing singularities and gauge shocks. For focusing
singularities, I extend the original analysis of Bona et. al and show that both
marginal and strong singularity avoidance can be obtained for certain types of
behavior of the slicing condition as the lapse approaches zero. For the case of
gauge shocks, I re-derive a condition found previously that eliminates them.
Unfortunately, such a condition limits considerably the type of slicings
allowed. However, useful slicing conditions can still be found if one asks for
this condition to be satisfied only approximately. Such less restrictive
conditions include a particular member of the 1+log family, which in the past
has been found empirically to be extremely robust for both Brill wave and black
hole simulations.Comment: 11 pages, revtex4. Change in acknowledgment
A hyperbolic slicing condition adapted to Killing fields and densitized lapses
We study the properties of a modified version of the Bona-Masso family of
hyperbolic slicing conditions. This modified slicing condition has two very
important features: In the first place, it guarantees that if a spacetime is
static or stationary, and one starts the evolution in a coordinate system in
which the metric coefficients are already time independent, then they will
remain time independent during the subsequent evolution, {\em i.e.} the lapse
will not evolve and will therefore not drive the time lines away from the
Killing direction. Second, the modified condition is naturally adapted to the
use of a densitized lapse as a fundamental variable, which in turn makes it a
good candidate for a dynamic slicing condition that can be used in conjunction
with some recently proposed hyperbolic reformulations of the Einstein evolution
equations.Comment: 11 page
Formulations of the 3+1 evolution equations in curvilinear coordinates
Following Brown, in this paper we give an overview of how to modify standard
hyperbolic formulations of the 3+1 evolution equations of General Relativity in
such a way that all auxiliary quantities are true tensors, thus allowing for
these formulations to be used with curvilinear sets of coordinates such as
spherical or cylindrical coordinates. After considering the general case for
both the Nagy-Ortiz-Reula (NOR) and the Baumgarte-Shapiro-Shibata-Nakamura
(BSSN) formulations, we specialize to the case of spherical symmetry and also
discuss the issue of regularity at the origin. Finally, we show some numerical
examples of the modified BSSN formulation at work in spherical symmetry.Comment: 19 pages, 12 figure
Gauge conditions for long-term numerical black hole evolutions without excision
Numerical relativity has faced the problem that standard 3+1 simulations of
black hole spacetimes without singularity excision and with singularity
avoiding lapse and vanishing shift fail after an evolution time of around
30-40M due to the so-called slice stretching. We discuss lapse and shift
conditions for the non-excision case that effectively cure slice stretching and
allow run times of 1000M and more.Comment: 19 pages, 14 figures, REVTeX, Added a missing Acknowledgmen
Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system
Several numerical relativity groups are using a modified ADM formulation for
their simulations, which was developed by Nakamura et al (and widely cited as
Baumgarte-Shapiro-Shibata-Nakamura system). This so-called BSSN formulation is
shown to be more stable than the standard ADM formulation in many cases, and
there have been many attempts to explain why this re-formulation has such an
advantage. We try to explain the background mechanism of the BSSN equations by
using eigenvalue analysis of constraint propagation equations. This analysis
has been applied and has succeeded in explaining other systems in our series of
works. We derive the full set of the constraint propagation equations, and
study it in the flat background space-time. We carefully examine how the
replacements and adjustments in the equations change the propagation structure
of the constraints, i.e. whether violation of constraints (if it exists) will
decay or propagate away. We conclude that the better stability of the BSSN
system is obtained by their adjustments in the equations, and that the
combination of the adjustments is in a good balance, i.e. a lack of their
adjustments might fail to obtain the present stability. We further propose
other adjustments to the equations, which may offer more stable features than
the current BSSN equations.Comment: 10 pages, RevTeX4, added related discussion to gr-qc/0209106, the
version to appear in Phys. Rev.
An alternative approach to solving the Hamiltonian constraint
Solving Einstein's constraint equations for the construction of black hole
initial data requires handling the black hole singularity. Typically, this is
done either with the excision method, in which the black hole interior is
excised from the numerical grid, or with the puncture method, in which the
singular part of the conformal factor is expressed in terms of an analytical
background solution, and the Hamiltonian constraint is then solved for a
correction to the background solution that, usually, is assumed to be regular
everywhere. We discuss an alternative approach in which the Hamiltonian
constraint is solved for an inverse power of the conformal factor. This new
function remains finite everywhere, so that this approach requires neither
excision nor a split into background and correction. In particular, this method
can be used without modification even when the correction to the conformal
factor is singular itself. We demonstrate this feature for rotating black holes
in the trumpet topology.Comment: 5 pages, 4 figures, matches version published in PR
- …