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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 295)
by jfaber＠einsteintoolkit.org 12 Mar '12

12 Mar '12

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

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 294)
by bcmsma＠astro.rit.edu 12 Mar '12

12 Mar '12

User: bmundim Date: 2012/03/12 03:37 PM Modified: / ET.tex Log: link to sec. 5.6.3 File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +3 -1 =================================================================== --- ET.tex 2012-03-12 20:31:14 UTC (rev 293) +++ ET.tex 2012-03-12 20:37:30 UTC (rev 294) @@ -2077,6 +2077,7 @@ matter field. \subsubsection{Gravitational Waves} +\label{sec:gw} One of the main goals of numerical relativity to date is modeling gravitational waveforms that may be used in template generation to help analyze @@ -2686,7 +2687,8 @@ calculated by the \codename{QuasiLocalMeasures} module, and were found to be $0.884 \mathrm{M}$ and $0.625 \mathrm{M}^{2}$, respectively. -Two modules are necessary to perform the waveform extraction. +Two modules are necessary to perform the waveform extraction +(see Sec.~\ref{sec:gw} for details). The first one, \codename{WeylScal4}, calculates the Weyl scalar $\Psi_4$ in term of the metric components and its derivatives; these were computed to be $4$-th order accurate in this example.

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 293)
by knarf＠cct.lsu.edu 12 Mar '12

12 Mar '12

User: knarf Date: 2012/03/12 03:31 PM Modified: / Log: use external pointer to https://svn.einsteintoolkit.org/cactus/EinsteinExamples/trunk/par/arXiv:111… Directory Changes: Directory: / ============ Property changes on: ___________________________________________________________________ Directory: /svn:externals/ ========================== - manifest https://svn.einsteintoolkit.org/manifest/trunk + manifest https://svn.einsteintoolkit.org/manifest/trunk EinsteinExamples https://svn.einsteintoolkit.org/cactus/EinsteinExamples/trunk/par/arXiv:111…

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 292)
by schnetter＠cct.lsu.edu 12 Mar '12

12 Mar '12

User: eschnett Date: 2012/03/12 02:17 PM Added: /figures/ withghost.fig, withghost.pdf Log: Add missing figure Directory Changes: Directory: /svn:mime-type/ ========================== + application/octet-stream File Changes: Directory: /figures/ ==================== File [added]: withghost.fig Delta lines: +163 -0 =================================================================== --- figures/withghost.fig (rev 0) +++ figures/withghost.fig 2012-03-12 19:17:05 UTC (rev 292) @@ -0,0 +1,163 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +2 1 0 2 0 7 50 0 -1 0.000 0 0 7 1 0 2 + 2 1 2.00 120.00 240.00 + 4650 5400 2700 4500 +2 1 0 2 0 7 50 0 -1 0.000 0 0 -1 0 0 2 + 2400 4200 6600 4200 +2 1 0 2 4 7 50 0 -1 0.000 0 0 7 0 0 2 + 2900 4000 2500 4400 +2 1 0 2 4 7 50 0 -1 0.000 0 0 7 0 0 2 + 2500 4000 2900 4400 +2 1 0 2 4 7 50 0 -1 0.000 0 0 7 0 0 2 + 3500 4000 3100 4400 +2 1 0 2 4 7 50 0 -1 0.000 0 0 7 0 0 2 + 3100 4000 3500 4400 +2 1 0 2 4 7 50 0 -1 0.000 0 0 7 0 0 2 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7 50 0 -1 0.000 0 0 7 0 0 2 + 8800 2800 8400 3200 +2 1 0 2 4 7 50 0 -1 0.000 0 0 7 0 0 2 + 6500 2800 6100 3200 +4 0 0 50 0 0 24 0.0000 4 330 1680 3675 2325 processor 0\001 +4 0 0 50 0 0 24 0.0000 4 255 660 900 3675 time\001 +4 0 0 50 0 0 24 0.0000 4 330 4245 4800 5700 boundary of physical domain\001 +4 0 0 50 0 0 24 0.0000 4 330 1650 6900 4725 ghostzones\001 +4 0 0 50 0 0 24 0.0000 4 255 690 7350 3825 copy\001 +4 0 0 50 0 0 24 0.0000 4 330 1680 9900 2325 processor 1\001 File [added]: withghost.pdf Delta lines: +0 -0 =================================================================== (Binary files differ) Property changes on: figures/withghost.pdf ___________________________________________________________________

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 291)
by eloisa.bentivegna＠aei.mpg.de 12 Mar '12

12 Mar '12

User: bentivegna Date: 2012/03/12 12:38 PM Modified: / ET.tex /examples/cosmology/figs/ err.pdf, kasner.pdf, kasner.py Log: Addressed first referee's comments to cosmology section. All the suggestions were incorporated. A few typos in the extrinsic curvature of Kasner have been fixed in the process. File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +14 -19 =================================================================== --- ET.tex 2012-03-12 17:12:48 UTC (rev 290) +++ ET.tex 2012-03-12 17:38:18 UTC (rev 291) @@ -3026,44 +3026,39 @@ problem for spacetimes with radically different topologies and global properties. In this section, we illustrate the evolution of an initial-data set representing a constant-$t$ section of a -spacetime from the Gowdy $T^3$ class~\cite{Gowdy:1971jh,New:1997me}. Models in -this class have the line element: +spacetime from the Gowdy $T^3$ class~\cite{Gowdy:1971jh,New:1997me}, namely +the Kasner model. This spacetime has the line element: \begin{equation} -\label{eq:gowdyT3} -ds^2=\tau^{-1/2}e^{\lambda/2}(-d\tau^2+dz^2)+\tau[e^P(dx+Qdy)^2+e^{-P}dy^2] -\end{equation} -defined on a 3-torus $-x_0 \leq x \leq x_0$, $-y_0 \leq y \leq y_0$, -$-z_0 \leq z \leq z_0$, with the functions $P$, $Q$ and $\lambda$ to be -determined by the Einstein equations. For $P=Q=\lambda=0$, a coordinate -transformation $t=4/3 \, \tau^{3/4}$ (plus a rescaling of the spatial -coordinates) casts the line element into the form: -\begin{equation} \label{eq:kasner} ds^2=-dt^2+t^{4/3}(dx^2+dy^2)+t^{-2/3}dz^2 \end{equation} -which represents the familiar Kasner spacetime for a homogeneous but -anisotropically expanding universe. In the 3+1 decomposition described -in Section~\ref{sec:ADMBase}, this reads: +defined on a 3-torus $-x_0 \leq x \leq x_0$, $-y_0 \leq y \leq y_0$, +$-z_0 \leq z \leq z_0$, with periodic boundary conditions. In the 3+1 decomposition +described in Section~\ref{sec:ADMBase}, this reads: \begin{widetext} \begin{eqnarray} \alpha(t) &=& 1 \\ \beta^i(t) &=& 0 \\ \gamma_{ij}(t) &=& {\rm diag}(t^{4/3},t^{4/3},t^{-2/3}) \\ -K_{ij}(t) &=& - {\rm diag}(\frac{2}{3} \, t^{4/3},\frac{2}{3} \, t^{4/3},\frac{1}{3} \, t^{-2/3}) +K_{ij}(t) &=& {\rm diag}(-\frac{2}{3} \, t^{1/3},-\frac{2}{3} \, t^{1/3},\frac{1}{3} \, t^{-5/3}) \end{eqnarray} \end{widetext} +This solution represents a vacuum, expanding universe with an homogeneous +but anisotropic metric tensor. In Figure~\ref{fig:kasner}, we show the full evolution of the $t=1$ slice of spacetime~\eref{eq:kasner}, along with the associated error for a sequence of -time resolutions. +time resolutions. We choose $x_0=y_0=z_0=5$ and spatial resolution equal to +$1$, and we run a set of four time resolutions equal to $[0.0125,0.025,0.05,0.1]$. \begin{figure} \includegraphics[width=0.9\textwidth]{kasner.pdf} \includegraphics[width=0.9\textwidth]{err.pdf} - \caption{Top: the evolution of a vacuum spacetime of the type~\eref{eq:gowdyT3}, - with $P=Q=\lambda=0$; the initial data are chosen as + \caption{Top: the evolution of a vacuum spacetime of the type~\eref{eq:kasner}; + the initial data are chosen as $\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$. - Bottom: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$; + Bottom: the numerical error for the sequence of four time resolutions $[0.0125,0.025,0.05,0.1]$; + the superscripts $n$ and $e$ indicate the numerical and the exact solutions respectively, and the errors are scaled according to the expectation for fourth-order convergence. \label{fig:kasner}} \end{figure} Directory: /examples/cosmology/figs/ ==================================== File [modified]: err.pdf Delta lines: +0 -0 =================================================================== (Binary files differ) File [modified]: kasner.pdf Delta lines: +0 -0 =================================================================== (Binary files differ) File [modified]: kasner.py Delta lines: +2 -0 =================================================================== --- examples/cosmology/figs/kasner.py 2012-03-12 17:12:48 UTC (rev 290) +++ examples/cosmology/figs/kasner.py 2012-03-12 17:38:18 UTC (rev 291) @@ -23,6 +23,7 @@ ax.plot(t[::5], ord2[::5], linestyle='none', marker='.', markersize=10, color='blue', label='$\gamma_{zz}$ (numerical)') ax.set_xlabel(r'$t$') +ax.set_ylabel(r'$\gamma_{xx}$') handles, labels = ax.get_legend_handles_labels() ax.set_yscale('log') @@ -44,6 +45,7 @@ axb.plot(t, (ord1-exa1)/16**3, linestyle='-', color='black', label='$(\gamma_{xx}^n-\gamma_{xx}^e)/16^3$') axb.set_xlabel(r'$t$') +axb.set_ylabel(r'Error in $\gamma_{xx}$') handles, labels = axb.get_legend_handles_labels() axb.legend(handles, labels, loc='best')

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 290)
by roland.haas＠physics.gatech.edu 12 Mar '12

12 Mar '12

User: rhaas Date: 2012/03/12 12:12 PM Modified: / ET.tex /figures/ rot180-grid.fig, rot180-grid.pdf Log: update rotation180 figure to make cyan points easier to see, update figure caption File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +1 -1 =================================================================== --- ET.tex 2012-03-12 16:40:21 UTC (rev 289) +++ ET.tex 2012-03-12 17:12:48 UTC (rev 290) @@ -2418,7 +2418,7 @@ consisting of $5\times3$ points (small, dark, filled circles in the upper half), \codename{CarpetRegrid2} enforced the the $\pi$-symmetry by adding the $2\times3$ block of refined points - in the lower half (small, light, filled circles). The open + in the lower half (small, light, slashed circles). The open circles are boundary and ghost points maintained by \codename{Carpet}. The arrow demonstrates how the $\pi$-symmetry fills in a boundary point.} Directory: /figures/ ==================== File [modified]: rot180-grid.fig Delta lines: +21 -9 =================================================================== --- figures/rot180-grid.fig 2012-03-12 16:40:21 UTC (rev 289) +++ figures/rot180-grid.fig 2012-03-12 17:12:48 UTC (rev 290) @@ -1,4 +1,4 @@ -#FIG 3.2 Produced by xfig version 3.2.5b +#FIG 3.2 Produced by xfig version 3.2.5a Landscape Center Inches @@ -56,16 +56,12 @@ 1 3 0 4 0 0 49 -1 20 0.000 1 0.0000 4800 1200 150 150 4800 1200 4950 1200 1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 3600 9600 150 150 3600 9600 3600 9750 1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 3600 8400 150 150 3600 8400 3600 8550 -1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 4800 8400 150 150 4800 8400 4800 8550 -1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 4800 9600 150 150 4800 9600 4800 9750 1 3 0 4 0 0 49 -1 20 0.000 1 0.0000 6000 0 150 150 6000 0 6150 0 1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 6000 150 150 7200 6000 7200 6150 1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 7200 150 150 7200 7200 7200 7350 1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 8400 150 150 7200 8400 7200 8550 1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 7200 9600 150 150 7200 9600 7350 9600 1 3 0 4 0 7 49 -1 20 0.000 1 0.0000 6000 6000 150 150 6000 6000 6000 6150 -1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 6000 9600 150 150 6000 9600 6150 9600 -1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 6000 8400 150 150 6000 8400 6000 8550 1 3 0 4 0 0 49 -1 20 0.000 1 0.0000 7200 0 150 150 7200 0 7200 150 1 3 0 4 0 0 49 -1 20 0.000 1 0.0000 4800 0 150 150 4800 0 4950 0 1 3 0 4 4 4 50 -1 20 0.000 1 0.0000 4800 0 335 335 4800 0 4950 300 @@ -73,9 +69,13 @@ 1 3 0 4 4 4 50 -1 20 0.000 1 0.0000 4800 2400 335 335 4800 2400 4950 2700 1 3 0 4 4 4 50 -1 20 0.000 1 0.0000 7200 2400 335 335 7200 2400 7350 2700 1 3 0 4 4 4 50 -1 20 0.000 1 0.0000 9600 2400 335 335 9600 2400 9750 2700 -1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 6000 7200 150 150 6000 7200 6150 7200 1 3 0 4 4 4 50 -1 20 0.000 1 0.0000 4800 7200 335 335 4800 7200 4950 7500 -1 3 0 4 0 16 49 -1 20 0.000 1 0.0000 4800 7200 150 150 4800 7200 4800 7350 +1 3 0 4 0 3 49 -1 20 0.000 1 0.0000 4800 7200 150 150 4800 7200 4950 7200 +1 3 0 4 0 3 49 -1 20 0.000 1 0.0000 4800 8400 150 150 4800 8400 4800 8550 +1 3 0 4 0 3 49 -1 20 0.000 1 0.0000 4800 9600 150 150 4800 9600 4800 9750 +1 3 0 4 0 3 49 -1 20 0.000 1 0.0000 6000 9600 150 150 6000 9600 6150 9600 +1 3 0 4 0 3 49 -1 20 0.000 1 0.0000 6000 8400 150 150 6000 8400 6000 8550 +1 3 0 4 0 3 49 -1 20 0.000 1 0.0000 6000 7200 150 150 6000 7200 6150 7200 2 1 1 4 0 16 51 -1 -1 10.000 0 0 -1 1 0 2 1 1 1.00 60.00 120.00 2400 4800 12000 4800 @@ -88,9 +88,21 @@ 2 1 0 4 0 16 49 -1 20 0.000 0 0 -1 1 0 2 1 1 4.00 240.00 480.00 4800 11550 4800 9900 +2 1 0 4 0 7 48 -1 44 0.000 0 0 -1 0 0 3 + 4697 7097 4800 7200 4903 7303 +2 1 0 4 0 7 48 -1 44 0.000 0 0 -1 0 0 3 + 5897 7097 6000 7200 6103 7303 +2 1 0 4 0 7 48 -1 44 0.000 0 0 -1 0 0 3 + 5897 8297 6000 8400 6103 8503 +2 1 0 4 0 7 48 -1 44 0.000 0 0 -1 0 0 3 + 5897 9497 6000 9600 6103 9703 +2 1 0 4 0 7 48 -1 44 0.000 0 0 -1 0 0 3 + 4697 9497 4800 9600 4903 9703 +2 1 0 4 0 7 48 -1 44 0.000 0 0 -1 0 0 3 + 4697 8297 4800 8400 4903 8503 3 0 1 4 0 7 48 -1 -1 10.000 0 1 0 3 2 0 4.00 180.00 480.00 6000 9600 2400 4800 3600 0 0.000 1.000 0.000 -4 1 0 49 -1 0 48 0.0000 4 735 1635 450 5700 y = 0\001 -4 1 0 49 -1 0 48 0.0000 4 555 1635 4800 12150 x = 0\001 +4 1 0 49 -1 0 48 0.0000 4 735 1650 450 5700 y = 0\001 +4 1 0 49 -1 0 48 0.0000 4 570 1650 4800 12150 x = 0\001 File [modified]: rot180-grid.pdf Delta lines: +0 -0 =================================================================== (Binary files differ)

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 289)
by jfaber＠einsteintoolkit.org 12 Mar '12

12 Mar '12

User: jfaber Date: 2012/03/12 11:40 AM Modified: / ET.tex Log: Notation table margin fixed other todo's handled, except for overall style review File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +15 -15 =================================================================== --- ET.tex 2012-03-12 16:35:53 UTC (rev 288) +++ ET.tex 2012-03-12 16:40:21 UTC (rev 289) @@ -490,11 +490,11 @@ step is called \emph{regridding}. Is most simulations using Carpet the sizes of refinement levels are -predescribed by the user, while their locations are adapted according -to the locations of interesting features such as black holes or stars. -In addition, refinement levels are often activated is disabled depending -on simulation dynamics, such as the formation of a common horizon, or -an increasing stellar density due to collaps. +predetermined by the user, while their locations are calculated adaptively +to track the locations of interesting features such as black holes or stars. +In addition, refinement levels may be disabled or activated depending +on the dynamics of a particular simulation, responding, e.g., to events such as the formation of a common horizon or +an increasing stellar density due to collapse, respectively. Since a finer grid spacing also requires smaller time steps for hyperbolic problems, the finer grids perform multiple time steps for @@ -696,12 +696,12 @@ \label{table:vars} {\centering \begin{tabular}{llll} -Symbol & Quantity & Equation & ET thorn::group \\ +Symbol & Quantity & Eqn. & ET thorn::group \\ \hline $G_{\mu\nu}$ & Einstein Tensor & \protect\ref{eq:einstein} & N/A\\ $T_{\mu\nu}$ & Stress-Energy Tensor & \protect\ref{eq:Tmunu} & TmunuBase::stress\_energy\_tensor\\ $g_{\mu\nu}$ & Spacetime 4-metric & \protect\ref{eq:adm} & N/A\\ -$F_{\mu\nu},~(^*F^{\mu\nu})$ & (Dual) Faraday tensor & \protect\ref{eq:Bi} & N/A\\ +$F_{\mu\nu}$ & Faraday tensor & \protect\ref{eq:Bi} & N/A\\ $u^\mu$ & 4-velocity & N/A & N/A\\ $\gamma_{ij}$ & Spatial 3-metric & \protect\ref{eq:adm} & ADMBase::metric\\ $\alpha$ & Lapse function & \protect\ref{eq:adm} & ADMBase::lapse\\ @@ -717,7 +717,7 @@ $\psi$, ($\phi$) & (Logarithmic) conformal factor & \protect\ref{eq:confpsi},\protect\ref{eq:phi} & (ML\_BSSN/ML\_log\_confac) \\ $\tilde{\gamma}_{ij}$ & Conformal 3-metric & \protect\ref{eq:tildegamma} & ML\_BSSN/ML\_metric \\ $K$ & Trace of extrinsic curvature & \protect\ref{eq:trK} & ML\_BSSN/ML\_trace\_curv \\ -$\tilde{A}_{ij}$ & Conformal traceless extrinsic curvature & \protect\ref{eq:Atij} & ML\_BSSN/ML\_curv \\ +$\tilde{A}_{ij}$ & Conformal extrinsic curvature & \protect\ref{eq:Atij} & ML\_BSSN/ML\_curv \\ $\tilde{\Gamma}^i$ & Conformal connection & \protect\ref{eq:Gammai} & ML\_BSSN/ML\_Gamma \\ $D$ & Conservative density & \protect\ref{eq:p2c1} & GRHydro::dens\\ $S^i$ & Conservative momentum & \protect\ref{eq:p2c2} & GRHydro::scon\\ @@ -1521,23 +1521,23 @@ density, pressure, and velocity. Wherever derivatives of hydrodynamic terms appear in the evolution equations for the conserved variables, they are restricted to appear only inside divergence terms -(referred to as fluxes) and never in the source terms. By calculating fluxes at cell faces, we may -obtain a consistent description of the inter-cell values using -reconstruction techniques that account for the fact that hydrodynamic -variables are not smooth and may not be finite differenced accurately. +(referred to as fluxes) and never in the source terms. By calculating fluxes between pairs of neighboring points where field and hydrodynamic terms are evaluated, we +obtain a consistent description of time evolution using HRSC +reconstruction techniques that accounts for the fact that hydrodynamic +variables are not smooth and may not be finite differenced accurately \footnote{While the \codename{Carpet} AMR driver uses a so-called ``vertex-centered'' approach, \codename{GRHydro} is sufficiently general in its techniques that it may be used with either ``cell-centered'' or ``vertex-centered'' grid infrastructures; see, e.g., \protect\cite{Baumgarte:2010nu} for a review of these methods}. All other source terms in the evolution equations may contain only the hydrodynamic variables themselves and the metric variables and derivatives of the latter, since the metric must formally be smooth and thus differentiable using finite differencing techniques. Summarizing these methods briefly, the following stages occur every timestep: \begin{itemize} -\item The primitive variables are ``reconstructed'' at cell faces +\item The primitive variables are ``reconstructed'' between neighboring points using shock-capturing techniques, with total variation diminishing (TVD), piecewise parabolic (PPM), and essentially non-oscillatory (ENO) methods currently implemented. -\item A Riemann problem is solved at each cell face using an +\item A Riemann problem is solved to generate the fluxes corresponding to the reconstructed values using an approximate solver. Currently implemented versions include HLLE (Harten-Lax-van Leer-Einfeldt), Roe, and Marquina solvers. -\item The conserved variables are advanced one timestep, and used to +\item Using the computed flux and source terms, the conserved variables are advanced one timestep, and used to recalculate the new values of the primitive variables. \end{itemize} We discuss the GRHD formalism, the stages within a timestep, and the

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 288)
by diener＠cct.lsu.edu 12 Mar '12

12 Mar '12

User: diener Date: 2012/03/12 11:35 AM Modified: / ET.tex Log: Add a sentence trying to address the referee's comment on boundary conditions. File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +8 -0 =================================================================== --- ET.tex 2012-03-12 16:09:44 UTC (rev 287) +++ ET.tex 2012-03-12 16:35:53 UTC (rev 288) @@ -1492,7 +1492,15 @@ observable, propagate faster, with a speed up to $\sqrt{2}$ for the gauge conditions used in \codename{McLachlan}. +Unfortunately more advanced radiative and/or constraint preserving boundary +conditions are not available in the Einstein Toolkit at the present time. +However, {\tt Cactus} is a flexible and extensible framework and supports data +structures (such as lower dimensional grid arrays) that could be used for +implementing more advanced outer boundary conditions. It will be the +requirements and needs of the numerical relativity community that decides +which enhancements and improvements will be added to the toolkit. + \subsection{Hydrodynamics Evolution} \label{sec:GRHydro}

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 287)
by schnetter＠cct.lsu.edu 12 Mar '12

12 Mar '12

User: eschnett Date: 2012/03/12 11:09 AM Modified: / ET.tex Log: Describe ghost zones File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +28 -10 =================================================================== --- ET.tex 2012-03-12 15:57:39 UTC (rev 286) +++ ET.tex 2012-03-12 16:09:44 UTC (rev 287) @@ -447,7 +447,23 @@ with parallelization, time evolution, or mesh refinement. The information provided in the interface declarations of the individual components allows a highly efficient execution of the combined -program. +program. Cactus's parallelization paradigm is based on a spatial +domain decomposition, and is briefly explained in figure +\ref{fig:ghosts}. +\begin{figure} + % Figure taken from Cactus users' guide + \centering + \includegraphics[width=0.7\textwidth]{withghost} + \caption{Cactus employs spatial domain decomposition to distribute + workload and storage across processors. It stores \emph{ghost + zones} (additional, ``dummy'' grid points, here shown as bold + and blue crosses) at inter-process boundaries to allow evaluating + computational stencils near these boundaries. After modifying + data, these ghost zones need to be \emph{synchronized}, which + requires inter-processor communication. This is handled by a + special \emph{driver} component (see main text).} + \label{fig:ghosts} +\end{figure} The Einstein Toolkit offers two drivers, \codename{PUGH} and {\tt Carpet}. \codename{PUGH} provides domains consisting of a uniform @@ -537,8 +553,8 @@ \small \centering \begin{tabular}{lll|rrr} - Name & Architecture (CPU) & Interconnect & nodes & cores/node & CPU - freq. \\\hline + Name & Architecture (CPU) & Interconnect & nodes & cores/node & + CPU freq. \\\hline Franklin (NERSC) & Cray XT4 (AMD) & SeaStar2 & 8502 & 4 & 2.3 GHz \\ HLRB II (LRZ Munich) & SGI Altix (Itanium) & NUMAlink & 1 & 9728 @@ -2265,8 +2281,7 @@ \codename{Cartoon2D} allows fully three dimensional codes to be used in axisymmetric problems by evolving a slice in the $y=0$ plane and using the rotational symmetry to populate -ghost points -\todo{ES: this is the first mention of ``ghost''} +boundary points off the plane (see Figure~\ref{fig:cartoon-plane}). \begin{figure}[htbp] \begin{center} @@ -2285,7 +2300,7 @@ \label{fig:cartoon-plane} \end{figure} -In applying symmetries to populate ghost zones, the +In applying symmetries to populate boundary zones, the transformation properties of tensorial quantities (including tensor densities and non-tensors such as Christoffel symbols) are correctly taken into account, just as they are in the interpolation routines present in {\tt Cactus}. @@ -2384,11 +2399,14 @@ % algorithm described below doesn't know about buffer points -- % these are handled before and afterwards.) \caption{Example of a grid layout created by - \codename{CarpetRegrid2}. In this example we use one boundary - point and one ghost point, as well as - \codename{RotatingSymmetry180}. This figure shows two refinement + \codename{CarpetRegrid2}. This figure shows two refinement levels, a coarse (big red circles) and a fine one (small black - circles). Starting from a user-specified refined region + circles). In this example we use one boundary point and one + ghost point, as well as \codename{RotatingSymmetry180}. The + boundary points are filled by the symmetry condition, the ghost + points are filled via interpolation from the coarse + grid.\newline + Starting from a user-specified refined region consisting of $5\times3$ points (small, dark, filled circles in the upper half), \codename{CarpetRegrid2} enforced the the $\pi$-symmetry by adding the $2\times3$ block of refined points

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 286)
by ian.hinder＠aei.mpg.de 12 Mar '12

12 Mar '12

User: hinder Date: 2012/03/12 10:57 AM Modified: / ET.tex Log: ET.tex: Rework discussion of NRAR error criteria according to referee comments File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +16 -12 =================================================================== --- ET.tex 2012-03-12 15:56:23 UTC (rev 285) +++ ET.tex 2012-03-12 15:57:39 UTC (rev 286) @@ -2728,18 +2728,22 @@ %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 $\omega=0.2/\mathrm{M}$. 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 of -$\omega=0.2/\mathrm{M}$. From the plot, we assert that the phase error between the -higher and high resolutions and the one between high and medium-high -resolutions satisfies 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 does 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. +wave frequency reaches $\omega=0.2/\mathrm{M}$. This reference time +has been suggested, for example in the NRAR\cite{NRAR:web} +collaboration, as a useful point at which errors can be compared +between codes. The suggestion is that the accumulated phase error +should not be larger than $0.05$ radians at this reference time, and +that this is roughly the scale of accuracy which might be useful for +comparisons with analytic approximation methods. From the plot, we +assert that the phase difference between the higher and high +resolutions and the one between high and medium-high resolutions +satisfies this criterion, while the phase difference between the +medium-high and medium resolutions barely satisfies the criterion; and +the one between medium and low resolutions does not. We conclude then +that the three highest resolution runs do have the suggested accuracy. +While the waveform shown here is too short for comparison with +analytic methods, we illustrate here that these sorts of comparisons +can be made using the toolkit. \begin{figure} \includegraphics[width=0.45\textwidth]{tracks}

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