[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 178)
jfaber at einsteintoolkit.org
jfaber at einsteintoolkit.org
Tue Nov 8 11:31:14 CST 2011
User: jfaber
Date: 2011/11/08 11:31 AM
Modified:
/
ET.tex
Log:
BH examples sections edited, and some of RIT's acknowledgments added
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +69 -64
===================================================================
--- ET.tex 2011-11-08 04:25:03 UTC (rev 177)
+++ ET.tex 2011-11-08 17:31:14 UTC (rev 178)
@@ -2276,14 +2276,15 @@
\label{fig:rot180-grid}
\end{figure}
-\section{Examples\pages{0.5 Frank}}
-\todo{Will be written once we know which examples we have}
+\section{Examples}
+\todo{Update if necessary}
To demonstrate the properties of the code and its capabilities, we have used it to simulate common astrophysical configurations of interest. Given the community-oriented direction of the project, the parameter files required to launch these simulations and a host of others are included and documented in the code releases, along with the datafiles produced by a representative set of simulation parameters to allow for code validation and confirmation of correct code performance on new platforms and architetures. As part of the internal validation process,
nightly builds are checked against a set of benchmarks to ensure that consistent results are generated with the inclusion of all new commits to the code.
The performance of the Toolkit for vacuum configurations is demonstrated through evolutions of single, rotating BHs and the merger of binary black hole configurations (sections~\ref{sec:1bh-example} and \ref{sec:bbh-example}, respectively). Linear oscillations about equilibirum for an isolated NS are discussed in section~\ref{sec:tov_oscillations}, and the collapse of a NS to a BH, including dynamical formation of a horizon, in section~\ref{sec:collapse_example}. Finally, to show a less traditional application of the code, we show its ability to perform cosmological simulations by evolving a Kasner spacetime (see section~\ref{sec:cosmology}).
-\subsection{Spinning BH\pages{2 Peter}}
+
+\subsection{Spinning BH}
\label{sec:1bh-example}
As a first example we perform simulations of a single distorted rotating BH.
We use \codename{TwoPunctures} to set up initial data for a single
@@ -2339,8 +2340,8 @@
\begin{figure}
\includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/waves_l4}
\includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/waves_l4_conv}
- \caption{The right plot shows the real part of the extracted
- $\ell =4, m=0$ mode of $\Psi_4$ as function of time from the high
+ \caption{Real part of the extracted
+ $\ell =4, m=0$ mode of $\Psi_4$ as function of time (left plot) for the high
(solid red curve), medium (long dashed green curve) and low (short
dashed blue
curve) resolution runs. The extraction was done at $R=30M$. The left
@@ -2363,21 +2364,21 @@
to perfect fourth order convergence.
In addition to the modes shown in Figure~\ref{fig:kerr_waves} and
-\ref{fig:kerr_waves_l4} we can mention that the extracted $\ell =4, m=4$ mode
+\ref{fig:kerr_waves_l4} we note that the extracted $\ell =4, m=4$ mode
is non-zero due to truncation error, but shows fourth order convergence to
zero with resolution (this mode is not present in the initial data and is not
excited during the evolution). Other modes are zero to round-off due to
symmetries at all resolutions.
-Since there is non-trivial gravitational wave content in the initial data
-the mass of the BH changes when evolved. In Figure~\ref{fig:ah_mass}
+Since there is non-trivial gravitational wave content in the initial data,
+the mass of the BH changes during its evolution. In figure~\ref{fig:ah_mass}
we show in the left plot the irreducible mass as calculated by
\codename{AHFinderDirect} as function of time at the low (red solid curve),
medium (green long dashed curve) and high (blue short dashed curve) resolutions.
\begin{figure}
\includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/ah_mass}
\includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/ah_mass_conv}
- \caption{The right plot shows the irreducible mass of the apparent horizon
+ \caption{The left plot shows the irreducible mass of the apparent horizon
as function of time at low (red solid curve), medium (green long dashed curve)
and high (blue short dashed curve) resolutions. The inset is a zoom in on the
$y$-axis to more clearly show the differences between the resolutions. The
@@ -2397,10 +2398,10 @@
(green long dashed curved) as well as the scaled difference between the
high and medium resolutions for fourth order (blue short dashed curve) and
third order (pink dotted curve). The convergence is almost perfectly
-fourth order until $T=50M$ then better than fourth order until $T=60M$ and
+fourth order until $T=50M$, then better than fourth order until $T=60M$, and
finally between third order and fourth order for the remainder of the
-evolution. The lack of perfect fourth order convergence at late time must
-be due to non-convergent errors from the puncture being able to propagate
+evolution. The lack of perfect fourth-order convergence at late times may be attributed
+ to non-convergent errors from the puncture propagating
to the horizon location at the lowest resolution.
Finally, in Figure~\ref{fig:ah_mass_spin} we show the total
@@ -2424,13 +2425,12 @@
right plot of Figure~\ref{fig:ah_mass_spin} by the fact that the pink
(dotted) curve (the high resolution result scaled by a factor of $1.78$ for
second order convergence to the resolution of the medium resolution) and the
-green (long dashed) curve are on top of each other. The reason for the
-second order convergence in the spin lies in the fact that thorn
-\codename{QuasiLocalMeasures} uses an algorithm which is
-only second order accurate overall. The increase of about 0.22\% in the mass of the
+green (long dashed) curve are on top of each other. Since the
+\codename{QuasiLocalMeasures} thorn uses an algorithm which is
+only second-order accurate overall, this is the expected result. The increase of about 0.22\% in the mass of the
BH is caused solely by the increase in the irreducible mass.
-\subsection{BHB\pages{2 Bruno}}
+\subsection{BH Binary}
\label{sec:bbh-example}
%BCM: DONE:ID parameters, TP number of collocation points, grid structure,
%coordinate domain, symmetries explored, description of the tracks,
@@ -2439,21 +2439,23 @@
%analysis were performed, table with remnant AH parameters: final
%spin, energy-momentum radiated with error bars estimated by the
%difference between the values of the highest resolutions
-\BCM{TODO: list the possible reasons we are not observing
-the convergence rate expected.}
+%JF: We have convergence plots before, so I'm not sure this is required for this section too
+%\BCM{TODO: list the possible reasons we are not observing
+%the convergence rate expected.}
+
To demonstrate the performance in the code for a problem of wide
current scientific interest, we have evolved a non-spinning equal-mass
BH binary system.
-The initial data was prepared so that the binary would be
+The initial data represent a binary
in a quasi-circular orbit, with an initial separation chosen
-to be $r=6M$ so we could track the later inspiral,
+to be $r=6M$ so we may track the later inspiral,
plunge, merger and ring down phases of the binary evolution.
Table~\ref{table:BHB_ID} provides more details about
the initial binary parameters used to generate the initial data.
The \codename{TwoPunctures} module uses these initial parameters
to solve \ref{eq:twopunc_u}, the elliptic Hamiltonian constraint for
-the regular component of the conformal factor (see Sec.~\ref{sec:twopunctures}).
+the regular component of the conformal factor (see section~\ref{sec:twopunctures}).
The spectral solution for this example was
determined by using $[n_A,n_B,n_{\phi}]=[28,28,14]$ collocation
points, and, along with the Bowen-York analytic solution for the
@@ -2491,16 +2493,16 @@
decomposition takes place.
Figure~\ref{fig:tracks_waveform} shows the two puncture tracks
-throughout all phases of the binary evolution. Tracks are
+throughout all phases of the binary evolution,
provided by the \codename{PunctureTracker} module. In the same
plot we have recorded the intersection
of the apparent horizon $2$-surface with the $z=0$ plane
every time interval $t=10M$ during the evolution.
A common horizon is first observed at $t=116M$. These apparent
horizons were found by the \codename{AHFinderDirect} module and their
-radius and location information stored into a $2$-surface of
+radius and location information stored as a $2$-surface with
spherical topology by the \codename{SphericalSurface} module.
-The irreducible mass and dimensionless spin of the remnant were
+The irreducible mass and dimensionless spin of the merged BH were
calculated by the \codename{QuasiLocalMeasures} module,
and were found to be $0.647 M$ and $-0.243 M^{-2}$, respectively.
@@ -2509,9 +2511,9 @@
$\Psi_4$ in term of the metric components and its derivatives;
these were computed to be $4$-th order accurate in this example.
The second module, \codename{Multipole}, interpolates the
-Weyl scalars into spheres with centers and radii specified by
+Weyl scalars onto spheres with centers and radii specified by
the user, and performs a spherical harmonic multipole
-mode decomposition on these spherical surfaces.
+mode decomposition.
Figure~\ref{fig:tracks_waveform} shows the
real and imaginary parts of the ($l=2$, $m=2$) mode for
$\Psi_4$ extracted on a sphere centered at the origin at $R_{\rm obs} = 60M$.
@@ -2530,55 +2532,58 @@
In order to evaluate the convergence of the numerical
solution, we ran five simulations with different
resolutions, and focus our analysis on the convergence
-of the Weyl scalar $\Psi_4$ phase and amplitude.
+of the phase and amplitude of the Weyl scalar $\Psi_4$.
The mesh spacings adopted for the coarser grid in the
AMR hierarchy for these different runs were
$\{h_{\rm low},h_{\rm med},h_{\rm medh},h_{\rm high},h_{\rm higher}\}
=\{2.0M,1.5M,1.25M,1.0M,0.75M\}$, respectively, while
the finer grid spacings can be easily found by dividing
-them by $2^{\rm level}$. In the case of this example, this
-results in
+them by $2^k$ for the $k$th level of mesh refinement.For this example, we set
$\{h^f_{\rm low},h^f_{\rm med},h^f_{\rm medh},h^f_{\rm high},h^f_{\rm higher}\}
=\{3.125M,2.344M,1.953M,1.563M,1.172M\}\times 10^{-2}$ for the
finest grid in the different AMR hierarchies, respectively.
-\BCM{Convergence factor text should be moved elsewhere -- starts here.}
-We may define the convergence factor as the order of the Richardson extrapolation,
+%JF - we assume this in the rpevious section with an equation, and can probably skip it here.
+%\BCM{Convergence factor text should be moved elsewhere -- starts here.}
+%We may define the convergence factor as the order of the Richardson extrapolation,
%
-\begin{equation}
-Q(t) = \frac{||u^{h_{\rm low}}-u^{h_{\rm med}}||}
- {||u^{h_{\rm med}}-u^{h_{\rm high}}||},
-\end{equation}
-where $||\cdot||$ refers to an appropriate norm, and
-$h_{\rm low}$, $h_{\rm med}$ and $h_{\rm high}$ correspond to grid
-spacings of low, medium and high resolutions, respectively.
-\BCM{stops here}
+%\begin{equation}
+%Q(t) = \frac{||u^{h_{\rm low}}-u^{h_{\rm med}}||}
+ % {||u^{h_{\rm med}}-u^{h_{\rm high}}||},
+%\end{equation}
+%where $||\cdot||$ refers to an appropriate norm, and
+%$h_{\rm low}$, $h_{\rm med}$ and $h_{\rm high}$ correspond to grid
+%spacings of low, medium and high resolutions, respectively.
+%\BCM{stops here}
+
Here, we consider the phase $\phi(t)$ and
the amplitude $A(t)$ of the Weyl scalar $\Psi_4$ at
$R_{\rm obs}=60M$. In order to take differences between
the numerical values at two different grid resolutions, we use
-an $8$-th order accurate Lagrange operator~\footnote{
+an $8$-th order accurate Lagrange operator to interpolate the higher-accuracy finite difference solution
+into the immediately coarser grid.
We have experimented with $4$-th and $6$-th order as well,
to evaluate the level of noise these interpolations could
-potentially introduce. We did not observe any noticeable
-difference and we decided to use the higher order option.}
-to interpolate the higher-accuracy finite difference solution
-into the immediately coarser grid.
-\BCM{Move this elsewhere too -- starts here}
-Since the grid spacings
-does not follow a $2:1$ ratio, the convergence factor has
-a different functional form than the usual $2^p$ for a $p$-th accurate
-finite difference scheme:
+potentially introduce, but did not observe any noticeable
+difference and we report here on results from the higher-order option.
+
+%JF: Do we need this at all, if we don't show a traditional convergence plot?
+%\BCM{Move this elsewhere too -- starts here}
+%Since the grid spacings
+%does not follow a $2:1$ ratio, the convergence factor has
+%a different functional form than the usual $2^p$ for a $p$-th accurate
+%finite difference scheme:
%
-\begin{equation}
-Q = \frac{b^p - a^p}{a^p - 1},
-\end{equation}
-where $h_{\rm high}=h$, $h_{\rm med}=a\times h$ and
-$h_{\rm low} = b \times h$. In our case, $a$ and $b$
-are $1.25$ and $1.5$, respectively. A $4$-th order convergent
-scheme for our grid spacings would give then a convergence
-factor of $Q = 1.818$, while a $6$-th and $8$-th order one,
-$2.691$ and $3.965$, respectively.
-\BCM{stops here. Remember to amend the text appropriately}
+%\begin{equation}
+%Q = \frac{b^p - a^p}{a^p - 1},
+%\end{equation}
+%where $h_{\rm high}=h$, $h_{\rm med}=a\times h$ and
+%$h_{\rm low} = b \times h$. In our case, $a$ and $b$
+%are $1.25$ and $1.5$, respectively. A $4$-th order convergent
+%scheme for our grid spacings would give then a convergence
+%factor of $Q = 1.818$, while a $6$-th and $8$-th order one,
+%$2.691$ and $3.965$, respectively.
+%\BCM{stops here. Remember to amend the text appropriately}
+
In Figure~\ref{fig:amp_phs_convergence}, we show the convergence
of the amplitude and phase of the Weyl scalar by plotting the
logarithm of the absolute value of the differences between two levels
@@ -2593,7 +2598,7 @@
wave frequency reaches $\omega=0.2/M$. We follow a community standard, agreed
to over the course of the NRAR\cite{NRAR:web} collaboration, that constrains
the numerical resolution so that the accumulated phase error is not
-larger than $0.05$ radians at a gravitational wave frequency around
+larger than $0.05$ radians at a gravitational wave frequency of
$\omega=0.2/M$. From the plot, we can assert that the phase error between the
higher and high resolutions, and the one between high and medium-high
resolutions satisfy this criterion, while the phase error between
@@ -2963,11 +2968,11 @@
National Science Foundation under the grant numbers
0903973/0903782/0904015 (CIGR\@). Related grants contribute directly
and indirectly to the success of CIGR, including NSF OCI-0721915, NSF
-OCI-0725070, NSF OCI-0905046, and NSF OCI 0941653, and NSF
-AST-0855535. Results presented in this article were obtained through
+OCI-0725070, NSF OCI-0905046, and NSF OCI 0941653, NSF
+AST-0855535, and NASA 08-ATFP08-0093. Results presented in this article were obtained through
computations on the Louisiana Optical Network Initiative under
allocation loni\_cactus05 and loni\_numrel06, as well as on the NSF
-Teragrid under allocation TG-MCA02N014, DOE repository m152, HLRB at
+Teragrid under allocations TG-MCA02N014 and TG-PHY060027N, DOE repository m152, HLRB at
the LRZ, and Compute Canada project cfz-411-aa.
\section*{References}
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