User: knarf
Date: 2011/11/14 02:35 AM
Modified:
/
ET.tex
Log:
cosmetics in examples section
File Changes:
Directory: /
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File [modified]: ET.tex
Delta lines: +65 -83
===================================================================
--- ET.tex 2011-11-14 08:00:28 UTC (rev 206)
+++ ET.tex 2011-11-14 08:35:47 UTC (rev 207)
@@ -2208,7 +2208,7 @@
puncture of mass $M_{\mathrm{bh}}=1$ and dimensionless spin parameter
$a = S_{\mathrm{bh}}/M_{\mathrm{bh}}^2 = 0.7$. Evolution of the data is
performed by \codename{McLachlan}, apparent horizon finding by
-\codename{AHFinderDirect}, and gravitational wave extraction by
+\codename{AHFinderDirect} and gravitational wave extraction by
\codename{WeylScal4} and \codename{Multipole}. Additional analysis of the
horizons is done by \codename{QuasiLocalMeasures}. The runs were performed
with fixed mesh refinement provided by \codename{Carpet}, using 8 levels
@@ -2238,7 +2238,7 @@
curve), between the high and medium resolution runs
(dashed blue curve), and the scaled difference (for 4th order
convergence) between the medium and low resolution runs
- (dotten red curve) for the real part of the $\ell =2, m=0$ waveforms.}
+ (dotted red curve) for the real part of the $\ell =2, m=0$ waveforms.}
\label{fig:kerr_waves}
\end{figure}
In the top plot the black (solid) curve is the real part and the blue (dashed)
@@ -2276,26 +2276,26 @@
actually small differences visible between resolutions in the beginning of
the waveform. The bottom plot shows the convergence of the real part of the
$\ell =4, m=0$ mode (compare with the bottom plot in Figure~\ref{fig:kerr_waves})
-and shows that even though the amplitude is much smaller we still obtain close
-to perfect fourth order convergence.
+and demonstrates that even though the amplitude is much smaller we still obtain close
+to perfect fourth-order convergence.
In addition to the modes shown in Figure~\ref{fig:kerr_waves} and
\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
+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 during its evolution. In figure~\ref{fig:ah_mass}
+the mass of the BH changes during its evolution. In figure~\ref{fig:ah_mass},
we show in the top plot the irreducible mass as calculated by
-\codename{AHFinderDirect} as function of time at the high (black solid curve),
+\codename{AHFinderDirect} as a function of time at the high (black solid curve),
medium (blue dashed curve) and low (red dotted curve) resolutions.
\begin{figure}
\includegraphics[width=0.9\textwidth]{examples/kerr/figs/ah_mass}
\includegraphics[width=0.9\textwidth]{examples/kerr/figs/ah_mass_conv}
\caption{The top plot shows the irreducible mass of the apparent horizon
-as function of time at low (black solid curve), medium (blue dashed curve)
+as a function of time at low (black solid curve), medium (blue dashed curve)
and high (red dotted curve) resolutions. The inset is a zoom in on the
$y$-axis to more clearly show the differences between the resolutions. The
bottom plot shows the convergence of the irreducible mass. The black (solid)
@@ -2303,7 +2303,7 @@
the blue (dashed) curve shows the difference between the high and medium
resolution results. The red (dotted) and green (dash-dotted) show the
difference between the high and medium resolutions scaled according to
-fourth and third order convergence respectively.} \label{fig:ah_mass}
+fourth and third-order convergence respectively.} \label{fig:ah_mass}
\end{figure}
The inset shows in more detail the differences between the different
resolutions. The irreducible mass increases by about 0.3\% during the first
@@ -2312,10 +2312,10 @@
irreducible mass by the difference between the medium and low resolutions
(black solid curve), the difference between the high and medium resolutions
(blue dashed curved) as well as the scaled difference between the
-high and medium resolutions for fourth order (red dotted curve) and
-third order (green dash-dotted curve). The convergence is almost perfectly
-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
+high and medium resolutions for fourth-order (red dotted curve) and
+third-order (green dash-dotted curve). The convergence is almost perfectly
+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 times may be attributed
to non-convergent errors from the puncture propagating
to the horizon location at the lowest resolution.
@@ -2326,21 +2326,21 @@
\begin{figure}
\includegraphics[width=0.9\textwidth]{examples/kerr/figs/qlm_mass}
\includegraphics[width=0.9\textwidth]{examples/kerr/figs/qlm_spin}
- \caption{The top plot shows the total mass and the right plot shows the changing spin (i.e.\ $\Delta S=S(t)-S(t=0)$ of the BH as function of time.
+ \caption{The top plot shows the total mass and the right plot shows the changing spin (i.e.\ $\Delta S=S(t)-S(t=0)$ of the BH as a function of time.
In both plots the black (solid) curve is for high, blue (dashed) for medium
and red (dotted) for low resolution. In the right plot the green (dash-dotted)
-curve shows the high resolution result scaled for second order convergence with
+curve shows the high resolution result scaled for second-order convergence with
the medium resolution.}
\label{fig:ah_mass_spin}
\end{figure}
-In both cases the black (solid) curve is high, blue (dashed) is medium and
-red (dotted) is low resolution. Since the spacetime is
-axisymmetric the gravitational waves can not radiate angular momentum. Thus
+In both cases the black (solid) curve is for high, blue (dashed) for medium and
+red (dotted) for low resolution. Since the spacetime is
+axisymmetric the gravitational waves cannot radiate angular momentum. Thus
any change in the spin must be due to numerical error and $\Delta S$ should
-converge to zero with increasing resolution. This is clearly seen in the
-bottom plot of Figure~\ref{fig:ah_mass_spin} by the fact that the green
+converge to zero with increasing resolution. This is clearly shown in the
+bottom plot of Figure~\ref{fig:ah_mass_spin}; the green
(dash-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
+second-order convergence to the resolution of the medium resolution) and the
blue (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
@@ -2348,22 +2348,10 @@
\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,
-%AH and the thorns used to generate them, description of the waveform,
-%its parameters and thorns used, description of how the convergence
-%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
-
-%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
+To demonstrate the performance in the code for a current problem of wide
+scientific interest, we have evolved a non-spinning equal-mass
BH binary system.
-The initial data represent a binary
+The initial data represent a binary system
in a quasi-circular orbit, with an initial separation chosen
to be $r=6M$ so we may track the later inspiral,
plunge, merger and ring down phases of the binary evolution.
@@ -2380,24 +2368,25 @@
by the \codename{McLachlan} module.
\begin{table}[!ht]
-\caption{Initial data parameters for a non-spinning equal mass
-BH binary. The punctures are located on the
-$x$-axis at positions $x_1$ and $x_2$, with puncture bare
-mass parameters $m_1 = m_2 = m$, and momenta $\pm\vec p$.
-}
\label{table:BHB_ID}
+{\centering
\begin{tabular}{l|llllll}
Configuration & $x_1$ & $x_2$ & $p_x$ & $p_y$ & $m$ & $M_{\rm ADM}$ \\
\hline
QC3 & 3.0 & -3.0 & 0.0 & 0.13808 & 0.47656 & 0.984618 \\
-\end{tabular}
+\end{tabular}\\}
+\caption{Initial data parameters for a non-spinning equal mass
+BH binary. The punctures are located on the
+$x$-axis at positions $x_1$ and $x_2$, with puncture bare
+mass parameters $m_1 = m_2 = m$, and momenta $\pm\vec p$.
+}
\end{table}
The simulation domain spans the coordinate range
$[[x_{\rm min},x_{\rm max}],[y_{\rm min},y_{\rm max}],[z_{\rm min},z_{\rm max}]]
= [[0,120],[-120,120],[0,120]]$, where we have taken advantage of
-both the equatorial symmetry, implemented by the
-\codename{ReflectionSymmetry} module, and the $180\degree$ rotational
+both the equatorial symmetry (implemented by the
+\codename{ReflectionSymmetry} module) and the $180\degree$ rotational
symmetry around the $z$-axis, which we apply at the $x=0$ plane using the
\codename{RotatingSymmetry180} module.
\codename{Carpet} provides a hierarchy of refined grids centered at
@@ -2515,11 +2504,11 @@
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 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
-the medium-high and medium resolutions barely satisfies the criterion, and the
-one between medium and low resolutions do not. We conclude then
+$\omega=0.2/M$. From the plot, we assert that the phase error between the
+higher and high resolutions and the one between high and medium-high
+resolutions satisfies this criterion, while the phase error between
+the medium-high and medium resolutions barely satisfies the criterion; and the
+one between medium and low resolutions does not. We conclude then
that the three highest resolution runs do have sufficient resolution
to extract waveforms for use in the construction of analytic waveform
templates.
@@ -2584,8 +2573,8 @@
resulting in a resolution of $0.125\mathrm{M}$ ($0.25\mathrm{M}$,
$0.5\mathrm{M}$) across the entire star.
-In figure~\ref{fig:tov_rho_max} we show the evolution of the central density of
-the star, over an evolution time of $1300\mathrm{M}$ ($6.5\mathrm{ms}$). The
+In Figure~\ref{fig:tov_rho_max} we show the evolution of the central density of
+the star over an evolution time of $1300\mathrm{M}$ ($6.5\mathrm{ms}$). The
initial spike is due to the perturbation of the solution resulting from the
interpolation onto the evolution grid. The remaining oscillations are mainly
due to the interaction of the star and the artificial atmosphere and are
@@ -2602,14 +2591,14 @@
from the interaction of the stellar surface with the artificial atmosphere.}
\end{figure}
-In figure~\ref{fig:tov_mode_spectrum} we show the power spectral density (PSD)
+In Figure~\ref{fig:tov_mode_spectrum} we show the power spectral density (PSD)
of the central density oscillations computed from a full 3D relativistic
hydrodynamics simulation, compared to the corresponding frequencies as obtained
-with perturbative techniques, kindly provided to use by Kentaro Takami and
-computed using the method described in~\cite{Yoshida:1999vj}. The PSD was computed
+with perturbative techniques (kindly provided by Kentaro Takami and
+computed using the method described in~\cite{Yoshida:1999vj}). The PSD was computed
using the entire time series of the high-resolution run, by removing the linear
trend and averaging over Hanning windows overlapping half the signal length after
-padding the signal to five time its length. The agreement of the
+padding the signal to five times its length. The agreement of the
fundamental mode and first three overtone frequencies is clearly visible, but
are limited beyond this by the finite numerical resolution.
Higher overtones should be measurable with higher
@@ -2639,7 +2628,7 @@
Figure~\ref{fig:tov_ham_conv} shows the order of convergence of the Hamiltonian
constraint violation, using the three highest-resolution runs, at the stellar
-center and a coordinate radius of $r=5\mathrm{M}$ which is about half-way between the
+center and a coordinate radius of $r=5\mathrm{M}$ which is about half way between the
center and the surface. The observed convergence rate for most of the
simulation time lies between $1.4$ and $1.5$ at the center, and between $1.6$ and
$2$ at $r=5\mathrm{M}$, consistent with the expected data-dependent convergence
@@ -2654,9 +2643,8 @@
second order of the hydrodynamics evolution scheme. This is
expected because the scheme's convergence rate drops to first order
at extrema or shocks, like the stellar center or surface.
- Consequently, the observed convergence order about half-way between
- the stellar center and surface is higher than $1.5$, but most of
- the time below $2$.}
+ Consequently, the observed convergence order about half way between
+ the stellar center and surface is higher than $1.5$, but mostly below $2$.}
\end{figure}
\subsection{Neutron star collapse\pages{2 Christian, Roland}}
@@ -2665,23 +2653,20 @@
or in a binary, or with a smooth singularity free spacetime, as in the
case of the TOV star. The evolution codes in the toolkit are,
however, also able to handle the dynamic formation of a singularity,
-that is follow a neutron star collapse into a BH, and as a simple
-example of this process we study the collapse of a non-rotating TOV
+that is follow a neutron star collapse into a BH. As a simple
+example of this process, we study the collapse of a non-rotating TOV
star. We create initial data as in section~\ref{sec:tov_oscillations}
using $\rho_c=3.154\times10^{-3}$ and $K_{\mathrm{ID}} = 100$, $\Gamma
-= 2$ yielding a star model of gravitational mass $1.67\,M_\odot$, that
+= 2$, yielding a star model of gravitational mass $1.67\,M_\odot$, that
is at the onset of instability. As is common in such situations~(e.g.,
\cite{Baiotti:2005vi}), we trigger collapse by reducing the pressure
support after the initial data have been constructed by lowering the
polytropic constant $K_{\mathrm{ID}}$ from its initial value to
-$K = 0.98 \, K_{\mathrm{ID}} = 98$. To ensure that the pressure
-depleted configuration remains a solution of the Einstein constraint
+$K = 0.98 \, K_{\mathrm{ID}} = 98$. To ensure that the pressure-depleted
+configuration remains a solution of the Einstein constraint
equations~\eref{eqn:analysis_hamiltonian_constraint} in the presence
-of matter we rescale the rest mass density $\rho$ such that the total
-energy density $T_{nn}$
-%\todo{RH: unify notation of $\rho$}
-%JF: Fixed, see above discussion
-does not change
+of matter, we rescale the rest mass density $\rho$ such that the total
+energy density $T_{nn}$ does not change:
\begin{equation}
\rho' + K (\rho')^2 = \rho + K_{\mathrm{ID}} \rho^2.
\label{eqn:collapse_rho_rescaled}
@@ -2691,23 +2676,20 @@
$K$ accelerates the onset of collapse that would otherwise rely on
being triggered by numerical noise, which would not be guaranteed to
converge to a unique solution with increasing resolution. In order to
-resolve the star as well as push the outer boundary far enough away so
+resolve the star as well as to push the outer boundary far enough away (so
that the star and the numerical outer boundary are not in causal
-contact during the simulation, we employ a fixed mesh refinement
+contact during the simulation) we employ a fixed mesh refinement
scheme. The outermost box has a radius of $R_0 = 204.8\,M_\odot$ and
a resolution of $3.2\,M_\odot$ ($2.4\,M_\odot$, $1.6\,M_\odot$,
$0.6\,M_\odot$ for higher convergence levels). Around the star,
centered about the origin, we stack $5$ extra boxes of approximate size
$8\times2^\ell\,M_\odot$ for $0 \le \ell \le 4$, where the resolution
-on each finer
-% RH: there is a likely typo in the paramter file which creates boxes of
-% radii: 2M,4M,8M,13.6M(!),32M,64M respectively. Changing it to 16M
-% doesn't really do any good or harm.
+on each
level is twice that of the surrounding level. In order to resolve the large
density gradients developing during the collapse, two more levels with radii
-$4\,M_\odot$ and $2\,M_\odot$ are present inside the star. We use the PPM
-reconstruction method and the HLLE Riemann solver to obtain second
-order convergent results in smooth regions. Due to the presence of the
+$4\,M_\odot$ and $2\,M_\odot$ are placed inside the star. We use the PPM
+reconstruction method and the HLLE Riemann solver to obtain second-order
+convergent results in smooth regions. Due to the presence of the
density maximum at the center of the star and the non-smooth atmosphere at the
edge of the star, we expect the observed convergence rate to be somewhat lower
than second order, but higher than first order.
@@ -2769,18 +2751,18 @@
violation~\eref{eqn:analysis_hamiltonian_constraint} at the center of the
star for a run with resolution $h_i$.
Up to the time when the apparent horizon forms, the convergence order is
-$\approx 1.5$ as expected. At later times, the singularity forming at the
+an expected $\approx 1.5$. At later times, the singularity forming at the
center of the collapsing star renders a pointwise measurement of the
convergence factor at the center impossible.
\subsection{Cosmology}\label{sec:cosmology}
The Einstein Toolkit is not only designed to evolve compact-object
-spacetimes, but it is also capable of solving the initial-value
-problem for spacetimes with radically different topology and global
-properties. In the following we illustrate the evolution of an
+spacetimes, but also to solve the initial-value
+problem for spacetimes with radically different topologies and global
+properties. In this section, we illustrate the evolution of an
initial-data set representing a constant-$t$ section of a
-spacetime from the Gowdy $T^3$ class~\cite{Gowdy:1971jh,New:1997me}; models in
+spacetime from the Gowdy $T^3$ class~\cite{Gowdy:1971jh,New:1997me}. Models in
this class have the line element:
\begin{equation}
\label{eq:gowdyT3}
