[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 120)

[diener at cct.lsu.edu]

User: diener
Date: 2011/08/23 11:18 AM

Modified:
 /
  ET.tex

Log:
 More on waves and convergence for the distorted rotating black hole
 example.

File Changes:

Directory: /
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File [modified]: ET.tex
Delta lines: +54 -6
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--- ET.tex	2011-08-23 16:16:58 UTC (rev 119)
+++ ET.tex	2011-08-23 16:18:18 UTC (rev 120)
@@ -2254,26 +2254,74 @@
 resolution on the coarsest grid. The runs where performed using the tapering
 evolution scheme in \codename{Carpet} in order to avoid interpolation in
 time during prolongation. The initial data corresponds to a rotating black
-hole perturbed by an $\ell =2, m=0$ Brill wave and as such has a non-zero
+hole perturbed by a Brill wave and as such has a non-zero
 gravitational wave content. We evolved using 4th order finite differencing from
 $T=0M$ until the black hole had settled down to a stationary state at $T=120M$.
 
+Figure~\ref{fig:kerr_waves} shows the $\ell =2, m=0$ mode of $r\Psi_4$ 
+extracted at $R=30M$ and its convergence.
 \begin{figure}
  \includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/waves}
  \includegraphics[width=0.33\textwidth,angle=-90]{examples/kerr/figs/waves_conv}
  \caption{The right plot shows the extracted $\ell =2, m=0$ mode of $\Psi_4$
           as function of time from the high resolution run. The extraction was
-          done at $R=30M$. Shown is both the real (solid red line) and the
-          imaginary (dashed green line) part of the waveform. The left plot
+          done at $R=30M$. Shown is both the real (solid red curve) and the
+          imaginary (dashed green curve) part of the waveform. The left plot
           shows for the real part of the $\ell =2, m=0$ waveforms the
           difference between the medium and low resolution runs (solid red
-          line), the difference between the high and medium resolution runs
-          (dashed green line) as well as the scaled (for 4th order
+          curve), the difference between the high and medium resolution runs
+          (dashed green curve) as well as the scaled (for 4th order
           convergence) difference between the medium and low resolution runs
-          (dotted blue line).}
+          (dotted blue curve).}
  \label{fig:kerr_waves}
 \end{figure}
+In the left plot the red (solid) curve is the real part and the green (dashed)
+curve is the imaginary part of $r \Psi_4$ for the high resolution run. Curves
+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 
+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
+scaled for 4th order convergence. With the resolutions used here this factor is
+$\left (0.016^4-0.024^4\right )/\left ( 0.012^4-0.016^4\right) \approx 5.94$.
 
+Figure~\ref{fig:kerr_waves_l4} shows similar plots for the $\ell =4, m=0$ mode
+of $r\Psi_4$, again extracted at $R=30 M$.
+\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
+          (solid red curve), medium (dashed green curve) and low (dotted 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
+          convergence) difference between the medium and low resolution runs
+          (dotted 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
+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
+$\ell =4, m=0$ mode (compare with the right 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. 
+
+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
+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
+symmetries at all resolution. 
+
+
+
 \subsection{BHB\pages{2 Bruno}}
 \label{sec:bbh-example}
 %BCM: DONE:ID parameters, TP number of collocation points, grid structure,