6 research outputs found
A numerical study of the quasinormal mode excitation of Kerr black holes
We present numerical results from three-dimensional evolutions of scalar perturbations of Kerr black holes. Our simulations make use of a high-order accurate multi-block code which naturally allows for fixed adaptivity and smooth inner (excision) and outer boundaries. We focus on the quasinormal ringing phase, presenting a systematic method for extraction of the quasinormal mode frequencies and amplitudes and comparing our results against perturbation theory. The amplitude of each mode depends exponentially on the starting time of the quasinormal regime, which is not defined unambiguously. We show that this time-shift problem can be circumvented by looking at appropriately chosen relative mode amplitudes. From our simulations we extract the quasinormal frequencies and the relative and absolute amplitudes of corotating and counterrotating modes (including overtones in the corotating case). We study the dependence of these amplitudes on the shape of the initial perturbation, the angular dependence of the mode and the black hole spin, comparing against results from perturbation theory in the so-called asymptotic approximation. We also compare the quasinormal frequencies from our numerical simulations with predictions from perturbation theory, finding excellent agreement. Finally we study under what conditions the relative amplitude between given pairs of modes gets maximally excited and present a quantitative analysis of rotational mode-mode coupling. The main conclusions and techniques of our analysis are quite general and, as such, should be of interest in the study of ringdown gravitational waves produced by astrophysical gravitational wave sources
New, efficient, and accurate high order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions
We construct new, efficient, and accurate high-order finite differencing
operators which satisfy summation by parts. Since these operators are not
uniquely defined, we consider several optimization criteria: minimizing the
bandwidth, the truncation error on the boundary points, the spectral radius, or
a combination of these. We examine in detail a set of operators that are up to
tenth order accurate in the interior, and we surprisingly find that a
combination of these optimizations can improve the operators' spectral radius
and accuracy by orders of magnitude in certain cases. We also construct
high-order dissipation operators that are compatible with these new finite
difference operators and which are semi-definite with respect to the
appropriate summation by parts scalar product. We test the stability and
accuracy of these new difference and dissipation operators by evolving a
three-dimensional scalar wave equation on a spherical domain consisting of
seven blocks, each discretized with a structured grid, and connected through
penalty boundary conditions.Comment: 16 pages, 9 figures. The files with the coefficients for the
derivative and dissipation operators can be accessed by downloading the
source code for the document. The files are located in the "coeffs"
subdirector
Numerical relativity with characteristic evolution, using six angular patches
The characteristic approach to numerical relativity is a useful tool in
evolving gravitational systems. In the past this has been implemented using two
patches of stereographic angular coordinates. In other applications, a
six-patch angular coordinate system has proved effective. Here we investigate
the use of a six-patch system in characteristic numerical relativity, by
comparing an existing two-patch implementation (using second-order finite
differencing throughout) with a new six-patch implementation (using either
second- or fourth-order finite differencing for the angular derivatives). We
compare these different codes by monitoring the Einstein constraint equations,
numerically evaluated independently from the evolution. We find that, compared
to the (second-order) two-patch code at equivalent resolutions, the errors of
the second-order six-patch code are smaller by a factor of about 2, and the
errors of the fourth-order six-patch code are smaller by a factor of nearly 50.Comment: 12 pages, 5 figures, submitted to CQG (special NFNR issue
AMR, stability and higher accuracy
Efforts to achieve better accuracy in numerical relativity have so far
focused either on implementing second order accurate adaptive mesh refinement
or on defining higher order accurate differences and update schemes. Here, we
argue for the combination, that is a higher order accurate adaptive scheme.
This combines the power that adaptive gridding techniques provide to resolve
fine scales (in addition to a more efficient use of resources) together with
the higher accuracy furnished by higher order schemes when the solution is
adequately resolved. To define a convenient higher order adaptive mesh
refinement scheme, we discuss a few different modifications of the standard,
second order accurate approach of Berger and Oliger. Applying each of these
methods to a simple model problem, we find these options have unstable modes.
However, a novel approach to dealing with the grid boundaries introduced by the
adaptivity appears stable and quite promising for the use of high order
operators within an adaptive framework
Exact boundary conditions in numerical relativity using multiple grids: scalar field tests
Cauchy-Characteristic Matching (CCM), the combination of a central 3+1 Cauchy
code with an exterior characteristic code connected across a time-like
interface, is a promising technique for the generation and extraction of
gravitational waves. While it provides a tool for the exact specification of
boundary conditions for the Cauchy evolution, it also allows to follow
gravitational radiation all the way to infinity, where it is unambiguously
defined.
We present a new fourth order accurate finite difference CCM scheme for a
first order reduction of the wave equation around a Schwarzschild black hole in
axisymmetry. The matching at the interface between the Cauchy and the
characteristic regions is done by transfering appropriate characteristic/null
variables. Numerical experiments indicate that the algorithm is fourth order
convergent. As an application we reproduce the expected late-time tail decay
for the scalar field.Comment: 14 pages, 5 figures. Included changes suggested by referee
A multi-block infrastructure for three-dimensional time-dependent numerical relativity
We describe a generic infrastructure for time evolution simulations in numerical relativity using multiple grid patches. After a motivation of this approach, we discuss the relative advantages of global and patch-local tensor bases. We describe both our multi-patch infrastructure and our time evolution scheme, and comment on adaptive time integrators and parallelisation. We also describe various patch system topologies that provide spherical outer and/or multiple inner boundaries. We employ penalty inter-patch boundary conditions, and we demonstrate the stability and accuracy of our three-dimensional implementation. We solve both a scalar wave equation on a stationary rotating black hole background and the full Einstein equations. For the scalar wave equation, we compare the effects of global and patch-local tensor bases, different finite differencing operators, and the effect of artificial dissipation onto stability and accuracy. We show that multi-patch systems can directly compete with the so-called fixed mesh refinement approach; however, one can also combine both. For the Einstein equations, we show that using multiple grid patches with penalty boundary conditions leads to a robustly stable system. We also show long-term stable and accurate evolutions of a one-dimensional non-linear gauge wave. Finally, we evolve weak gravitational waves in three dimensions and extract accurate waveforms, taking advantage of the spherical shape of our grid lines. Comment of the Author: Invited papers on numerical relativity, related to the Banff International Research Station programme 16–21 April 2005 and the Newton Institute programme 8 August–23 December 200