User: jfaber Date: 2012/03/12 09:32 PM
Modified: / ET.tex
Log: Minor stylistic changes to examples. MAde sure to thank the refs.
File Changes:
Directory: / ============
File [modified]: ET.tex Delta lines: +23 -22 =================================================================== --- ET.tex 2012-03-12 20:37:30 UTC (rev 294) +++ ET.tex 2012-03-13 02:32:05 UTC (rev 295) @@ -2469,14 +2469,14 @@ evolution of the black hole is not influenced by the imperfect boundary condition. We performed runs at 3 different -resolutions: the low resolution was $0.024\mathrm{M} (3.072\mathrm{M})$, medium was -$0.016\mathrm{M} (2.048\mathrm{M})$ and high was $0.012\mathrm{M} (1.536\mathrm{M})$, where the numbers refer to the -resolution on the finest (coarsest) grid. Each refined grid had twice the +resolutions: the low resolution grid spacings were $0.024\mathrm{M}/3.072\mathrm{M}$, medium were +$0.016\mathrm{M}/2.048\mathrm{M}$ and high were $0.012\mathrm{M}/1.536\mathrm{M}$, where the numbers refer to the +resolution on the finest/coarsest grid. Each refined grid had twice the resolution and half the side length of the containing grid. The runs were performed using the tapering evolution scheme in \codename{Carpet} to avoid interpolation in time during prolongation. The initial data correspond to a rotating, stationary Kerr BH -perturbed by a Brill wave\cite{Brandt:1996si} and, as such, has a non-zero +perturbed by a Brill wave\cite{Brandt:1996si} and, as such, have a non-zero gravitational wave content. We evolved the BH using 4th-order finite differencing from $T=0\mathrm{M}$ until it had settled down to a stationary state at $T=120\mathrm{M}$.
@@ -2513,7 +2513,7 @@ waveforms while the blue (dashed) curve shows the absolute value of the difference between the high and medium resolution waveforms in a log-plot. The red (dotted) curve is the same as the blue (dashed) curve, except it is -scaled for 4th order convergence, demonstrating fourth-order convergence. +scaled for 4th order convergence, demonstrating we indeed achieve this. With the resolutions used here this factor is $\left (0.016^4-0.024^4\right )/\left ( 0.012^4-0.016^4\right) \approx 5.94$.
@@ -2663,15 +2663,15 @@ \codename{ReflectionSymmetry} and \codename{RotatingSymmetry180}). The outer radius is located at $R = 120\mathrm{M}$, where $\mathrm{M}$ is the initial ADM mass of the binary system. We performed runs at 5 different resolutions: -$3.125\mathrm{M}\times10^{-2} (2.0\mathrm{M})$, -$2.344\mathrm{M}\times10^{-2} (1.5\mathrm{M})$, -$1.953\mathrm{M}\times10^{-2} (1.25\mathrm{M})$, -$1.563\mathrm{M}\times10^{-2} (1.0\mathrm{M})$ and -$1.172\mathrm{M}\times10^{-2} (0.75\mathrm{M})$, where the numbers refer to -the resolution on the finest (coarsest) grid. Two set of moving boxes, each -centered on one of the black holes, are used, with the finest grid having side +$0.03125\mathrm{M}/2.0\mathrm{M}$, +$0.02344\mathrm{M}/1.5\mathrm{M}$, +$0.01953\mathrm{M}/1.25\mathrm{M}$, +$0.01563\mathrm{M}/1.0\mathrm{M}$ and +$0.01172\mathrm{M}/0.75\mathrm{M}$, where the numbers refer to +the resolution on the finest/coarsest grid. Two sets of moving boxes, each +centered on one of the black holes, are used, with the finest grid having a side length of $2\mathrm{M}$ and each coarser grid having twice the size and half -the resolution of the contained finer grid. +the resolution of the grid it contained.
Figure~\ref{fig:tracks_waveform} shows the two puncture tracks throughout all phases of the binary evolution, @@ -2683,7 +2683,7 @@ horizons were found by the \codename{AHFinderDirect} module and their radius and location information stored as a $2$-surface with spherical topology by the \codename{SphericalSurface} module. -The irreducible mass and angular momentum (spin) of the merged BH were +The irreducible mass and spin angular momentum of the merged BH were calculated by the \codename{QuasiLocalMeasures} module, and were found to be $0.884 \mathrm{M}$ and $0.625 \mathrm{M}^{2}$, respectively.
@@ -2838,9 +2838,9 @@ centered around the star at the origin, each doubling the resolution, with sizes of $240\mathrm{M}$, $120\mathrm{M}$, $60\mathrm{M}$ and $30\mathrm{M}$. We perform runs at 3 different -resolutions: the low resolution is $0.500\mathrm{M} (8.0\mathrm{M})$, medium was -$0.250\mathrm{M} (4.0\mathrm{M})$ and high was $0.125\mathrm{M} (2.0\mathrm{M})$, -where the numbers refer to the resolution on the finest (coarsest) grid. +resolutions: the low resolution is $0.500\mathrm{M}/8.0\mathrm{M}$, medium was +$0.250\mathrm{M}/4.0\mathrm{M}$ and high was $0.125\mathrm{M}/2.0\mathrm{M}$, +where the numbers refer to the resolution on the finest/coarsest grid. Each refined grid had twice the resolution and half the side length of the containing grid, with the finest grid completely covering the star.
@@ -2961,11 +2961,11 @@ region during collapse. The outer boundaries were placed at $R=204.8\mathrm{M}$. We perform runs at 4 different resolutions: from lowest to highest the resolution are -$0.025\mathrm{M} (3.2\mathrm{M})$, -$0.0188\mathrm{M} (2.4\mathrm{M})$, -$0.0125\mathrm{M} (1.6\mathrm{M})$ and -$4.67\times10^{-3}\mathrm{M} (0.6\mathrm{M})$, -where the numbers refer to the resolution on the finest (coarsest) grid. +$0.025\mathrm{M}/3.2\mathrm{M}$, +$0.0188\mathrm{M}/2.4\mathrm{M}$, +$0.0125\mathrm{M}/1.6\mathrm{M}$ and +$4.67\times10^{-3}\mathrm{M}/0.6\mathrm{M}$, +where the numbers refer to the resolution on the finest/coarsest grid.
We use the PPM reconstruction method and the HLLE Riemann solver (see @@ -3203,6 +3203,7 @@ We are also grateful to the large number of people who contributed to the Einstein Toolkit via ideas, code, documentation, and testing; without these contributions, this toolkit would not exist today. +We thank our referees for their diligence and their many helpful suggestions.
The Einstein Toolkit is directly supported by the National Science Foundation in the USA under the grant numbers