[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 122)

[diener at cct.lsu.edu]

User: diener
Date: 2011/08/23 02:30 PM

Modified:
 /
  ET.tex

Log:
 Add stuff about irreducible mass and convergence for the distorted
 rotating black hole example.

File Changes:

Directory: /
============

File [modified]: ET.tex
Delta lines: +44 -11
===================================================================
--- ET.tex	2011-08-23 19:30:04 UTC (rev 121)
+++ ET.tex	2011-08-23 19:30:54 UTC (rev 122)
@@ -2270,9 +2270,9 @@
           shows for the real part of the $\ell =2, m=0$ waveforms the
           difference between the medium and low resolution runs (solid red
           curve), the difference between the high and medium resolution runs
-          (dashed green curve) as well as the scaled (for 4th order
+          (long dashed green curve) as well as the scaled (for 4th order
           convergence) difference between the medium and low resolution runs
-          (dotted blue curve).}
+          (short dashed blue curve).}
  \label{fig:kerr_waves}
 \end{figure}
 In the left plot the red (solid) curve is the real part and the green (dashed)
@@ -2280,9 +2280,9 @@
 from the lower resolution are indistinguishable from the high resolution curve
 at this scale. In the right plot the red (solid) curve shows the absolut value
 of the difference between the real part of the medium and low resolution
-waveforms while the green (dashed) curve shows the aboslute value of the 
+waveforms while the green (long dashed) curve shows the aboslute value of the 
 difference between the high and medium resolution waveforms in a log-plot.
-The blue (dotted) curve is the same as the green (dashed) curve, except it is
+The blue (short dashed) curve is the same as the green (dashed) curve, except it is
 scaled for 4th order convergence. With the resolutions used here this factor is
 $\left (0.016^4-0.024^4\right )/\left ( 0.012^4-0.016^4\right) \approx 5.94$.
 
@@ -2293,19 +2293,20 @@
  \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
-          (solid red curve), medium (dashed green curve) and low (dotted blue
+          (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
           plot shows for the real part of the $\ell =4, m=0$ waveforms the
           difference between the medium and low resolution runs (solid red
           curve), the difference between the high and medium resolution runs
-          (dashed green curve) as well as the scaled (for 4th order
+          (long dashed green curve) as well as the scaled (for 4th order
           convergence) difference between the medium and low resolution runs
-          (dotted blue curve).}
+          (short dashed blue curve).}
  \label{fig:kerr_waves_l4}
 \end{figure}
 The left plot in this case shows only the real part of the extracted waveform
-but for all three resolutions (red solid curve is high, green dashed curve is
-medium and blue dotted curve is low resolution). Since the amplitude of this
+but for all three resolutions (red solid curve is high, green long dashed curve is
+medium and blue short dashed curve is low resolution). Since the amplitude of this
 mode is almost a factor of 20 smaller than the $\ell =2, m=0$ mode there are
 actually small differences visible between resolutions in the beginning of
 the waveform. The right plot shows the convergence of the real part of the
@@ -2317,11 +2318,43 @@
 \ref{fig:kerr_waves_l4} we can mention 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 roundoff due to
+excited during the evolution). Other modes are zero to roundoff due to
 symmetries at all resolution. 
 
+Since there is non-trivial gravitational wave content in the initial data
+the mass of the black hole changes when evolved. In Figure~\ref{fig:ah_mass}
+we show in the left plot the irreducible mass as calculates 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
+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 reolsutions. The
+right plot shows the convergence of the irreducible mass. The red (solid)
+curve shows the difference between the medium and low resolution results,
+the green (long dashed) curve shows the difference between the high and medium
+resolution results. The blue (short dashed) and pink (dotted) shows the 
+difference between the high and medium resolutions scaled according to
+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
+$40M$ of evolution and then remains constant (within numerical error) for the
+remainder of the evolution. The right plot shows the convergence of the
+irreducible mass by the difference between the medium and low resolutions 
+(red solid curve), the difference between the high and medium resolutions
+(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
+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
+to the horizon location at the lowest resolution.
 
-
 \subsection{BHB\pages{2 Bruno}}
 \label{sec:bbh-example}
 %BCM: DONE:ID parameters, TP number of collocation points, grid structure,