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: / ============
File [modified]: ET.tex Delta lines: +54 -6 =================================================================== --- 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,