User: bmundim
Date: 2011/10/03 10:56 PM
Modified:
/
ET.tex
Log:
A few more edits on BHB section.
File Changes:
Directory: /
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File [modified]: ET.tex
Delta lines: +97 -62
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--- ET.tex 2011-10-03 19:43:07 UTC (rev 143)
+++ ET.tex 2011-10-04 03:56:39 UTC (rev 144)
@@ -871,7 +871,7 @@
regions, convenient for hydrodynamics tests.
-\subsubsection{TwoPunctures: Binary Black Holes and extensions}
+\subsubsection{TwoPunctures: Binary Black Holes and extensions}\label{sec:twopunctures}
A substantial fraction of the published work on the components of the Einstein toolkit
involves the evolution of binary black hole systems.
The most widely used routine to generate initial data for these is the
@@ -900,7 +900,7 @@
where $m_i$ and $r_i$ are the mass of and distance to each BH, respectively, the Hamiltonian
constraint may be written
\begin{eqnarray*}
-\Delta u +\left[\frac{1}{8}\alpha^7K^{ij}K_{ij}\right](1+\alpha u)^{-7}
+\Delta u +\left[\frac{1}{8}\alpha^7K^{ij}K_{ij}\right](1+\alpha u)^{-7}\label{eq:twopunc_u}
\end{eqnarray*}
subject to the boundary condition $u\rightarrow 1$ as $r\rightarrow\infty$. In Cartesian
coordinates, the function $u$ is infinitely differentiable everywhere except the
@@ -2390,27 +2390,29 @@
%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,
+%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, table with remnant AH parameters: final
-spin, energy-momentum radiated with error bars estimated by the
-difference between the values of the highest resolutions.}
+the convergence rate expected.}
-This example regards a non-spinning equal mass black-hole binary.
-The initial data was prepared such that the binary would be
-in a quasi-circular orbital motion. The initial separation was chosen
-to be $r=6M$ in order to still be able to track the later inspiral,
-plunge, merger and ring down phases of the binary dynamics during
-its evolution. Table~\ref{table:BHB_ID} provides more details about
+To demonstrate the performance in the code for a problem of wide
+current scientific interest, we have evolved a non-spinning equal
+mass black-hole binary system.
+The initial data was prepared so that the binary would be
+in a quasi-circular orbit, with an initial separation chosen
+to be $r=6M$ so we could 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 spectrally the Hamiltonian constraint elliptic equation for
-the conformal factor regular part. The spectral solution in this example was
+to solve Eq.~\ref{eq:twopunc_u}, the elliptic Hamiltonian constraint for
+the regular component of the conformal factor (see Sec.~\ref{sec:twopunctures}).
+The spectral solution for this example was
determined by using $[n_A,n_B,n_{\phi}]=[28,28,14]$ collocation
-points. Along with the Bowen-York analytic solution for the
-momentum constraints, the spectral solution is then interpolated
-in the simulation domain, representing thus a constrained
-initial data $\{\gamma_{ij},K_{ij}\}$. Its evolution is performed
+points, and, along with the Bowen-York analytic solution for the
+momentum constraints, represents constrained GR
+initial data $\{\gamma_{ij},K_{ij}\}$. The evolution is performed
by the \codename{McLachlan} module.
\begin{table}[!ht]
@@ -2430,44 +2432,46 @@
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
-the symmetries of the problem. More specifically, $180\degree$ rotational
-symmetry around the $z$-axis is applied at the $x=0$ plane by the
-\codename{RotatingSymmetry180} module, and a reflection symmetry
-at the $z=0$ plane by the \codename{ReflectionSymmetry} module.
-\codename{Carpet} provides a hierarchy of refined grids centred at
-each puncture. In this example we used $7$ levels of refinement,
+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
+each puncture. Here, we used $7$ levels of refinement,
where the box edge coordinate lengths are given by
-$[128,32,16,8,4,2]$ in units of the binary total mass $M=1$.
+$[128,32,16,8,4,2]$ in units of the total binary mass, which is set to unity.
Note that overlapping boxes are automatically redefined by
\codename{Carpet} into one unique region before the domain
decomposition takes place.
-Figure~\ref{fig:tracks_waveform} shows the two punctures tracks
-throughout all phases of the binary dynamics. Tracks are
-provided there by the \codename{PunctureTracker} module. Also in the same
-plot we have recorded every $10M$ of evolution, the intersection
-of the apparent horizon $2$-surface with the $z=0$ plane.
+Figure~\ref{fig:tracks_waveform} shows the two puncture tracks
+throughout all phases of the binary evolution. Tracks are
+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 \codename{AHFinderDirect} module and their
+horizons were found by the \codename{AHFinderDirect} module and their
radius and location information stored into a $2$-surface of
-spherical topology by \codename{SphericalSurface} module.
+spherical topology by the \codename{SphericalSurface} module.
+The irreducible mass and dimensionless spin of the remnant were
+calculated by the \codename{QuasiLocalMeasures} module,
+and were found to be $0.647 M$ and $-0.243 M^{-2}$, respectively.
Two modules are necessary to perform the waveform extraction.
The first one, \codename{WeylScal4}, calculates the Weyl scalar
$\Psi_4$ in term of the metric components and its derivatives;
-that were taken to be $4$-th order accurate in this example.
+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
the user, and performs a spherical harmonic multipolar
mode decomposition on these spherical surfaces.
Figure~\ref{fig:tracks_waveform} shows the
real and imaginary parts of the ($l=2$, $m=2$) mode for
-$\Psi_4$ extracted in a sphere centred at the origin and with
-an observer radius located at $R_{\rm obs} = 60M$.
+$\Psi_4$ extracted on a sphere centered at the origin at $R_{\rm obs} = 60M$.
The number of grid points on the sphere was set to be
-$[n_{\theta},n_{\phi}]=[120,240]$, what gives an angular
-resolution in this case of the order of $2.6 \times 10^{-2}$
-radians,
+$[n_{\theta},n_{\phi}]=[120,240]$, which yields an angular
+resolution of $2.6 \times 10^{-2}$ radians,
and an error of the same order, since the surface integrals were
calculated by midpoint rule -- a first order accurate method.
%BCM: This is certainly not correct. Give a better
@@ -2478,17 +2482,21 @@
%at this extraction region ($\sim 6.25 \times 10^{-2}$).
In order to evaluate the convergence of the numerical
-solution, we ran three different simulations with different
-resolutions, and focused our analysis on the convergence
+solution, we ran five different simulations with different
+resolutions, and focus our analysis on the convergence
of the Weyl scalar $\Psi_4$ phase and amplitude.
The mesh spacings adopted for the coarser grid in the
-AMR hierarchy were
-$\{h_{\rm low},h_{\rm med},h_{\rm high}\}=\{1.5,1.25,1.0\}$,
-respectively, while the finer grid spacings are easily
-found by dividing them by $2^{\rm level}$. The convergence
-factor is an common quantity used to study the rate of
-convergence of the finite difference numerical solution, $u^h$,
-to the continuum one. It is usually defined as:
+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
+$\{h^f_{\rm low},h^f_{\rm med},h^f_{\rm medh},h^f_{\rm high},h^f_{\rm higher}\}
+=\{3.125M,2.344M,1.953M,1.563M,1.172M\}\times 10^{-2}$ 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,
%
\begin{equation}
Q(t) = \frac{||u^{h_{\rm low}}-u^{h_{\rm med}}||}
@@ -2497,7 +2505,8 @@
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.
-In this example, we used the phase $\phi(t)$ and
+\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
@@ -2506,10 +2515,12 @@
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 accurate finite difference solution
-into the immediately coarser grid. Since the grid spacings
+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 from the usual $2^p$ for a $p$-th accurate
+a different functional form than the usual $2^p$ for a $p$-th accurate
finite difference scheme:
%
\begin{equation}
@@ -2521,24 +2532,41 @@
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
of resolution. The differences clearly converge to zero as the resolution
-is increased. We also show that the most appropriate convergence factor
-seems to be the one corresponding to an $8$-th order accurate finite
-difference approximation.
+is increased.
+%BCM: Until we have clearer argument on why we have superconvergence,
+% I am commenting out the following sentence:
+%We also show that the most appropriate convergence factor
+%seems to be the one corresponding to an $8$-th order accurate finite
+%difference approximation.
+We also indicate on both plots the time at which the gravitational
+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
+$\omega=0.2/M$. 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
+that the three highest resolution runs do have sufficient resolution
+to extract waveforms for use in the construction of analytic waveform
+templates.
\begin{figure}
\includegraphics[width=0.45\textwidth]{examples/bbh/figs/tracks}
\includegraphics[width=0.45\textwidth]{examples/bbh/figs/mp_psi4_l2_m2_r60}
- \caption{On the left panel, we have plotted the tracks corresponding to
+ \caption{In the left panel, we plot the tracks corresponding to
the evolution of two punctures initially located on the $x$-axis at $x=\pm 3$.
-The solid blue line represents puncture 1, while the dashed red line the
+The solid blue line represents puncture 1, and the dashed red line
puncture 2. The circular dotted green lines are the intersections of the
-apparent horizons with the $z=0$ plane plotted every $10M$ of binary
-evolution. The common horizon arises at $t=116M$. On the right panel,
-we have plotted the real (solid blue line) and imaginary (dotted red line)
+apparent horizons with the $z=0$ plane plotted every $10M$ during the binary
+evolution. A common horizon appears at $t=116M$. In the right panel,
+we plot the real (solid blue line) and imaginary (dotted red line)
parts of the ($l=2$,$m=2$) mode of the Weyl scalar $\Psi_4$ as extracted
at an observer radius of $R_{\rm obs}=60M$.}
\label{fig:tracks_waveform}
@@ -2548,9 +2576,16 @@
\includegraphics[width=0.45\textwidth]{examples/bbh/figs/amp_convergence_all_8th}
\includegraphics[width=0.45\textwidth]{examples/bbh/figs/phase_convergence_all_8th}
\caption{Weyl scalar amplitude (left panel) and phase (right panel)
-convergence. The dotted vertical green line drawn approximately at $t=154M$
-indicates the evolution time in which the Weyl scalar frequency reaches
-$\omega=0.2/M$. in progress... }
+convergence. The long dashed red curves represent the difference between
+the medium and low-resolution runs. The short dashed orange curves show
+the difference between the medium-high and medium resolution runs. The
+dotted brown ones, the difference between high and medium-high resolutions,
+while the solid blue curves represent the difference between the higher
+and high resolution runs. The dotted vertical green line
+at $t=154M$ indicates the point during the evolution at which the Weyl
+scalar frequency reaches $\omega=0.2/M$. Observe that the three highest
+resolutions accumulate a phase error below the standard of $0.05$ radians
+required by the NRAR collaboration. }
\label{fig:amp_phs_convergence}
\end{figure}
