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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/examples/kerr/figs/ (Rev. 124)
by diener＠cct.lsu.edu 24 Aug '11

24 Aug '11

User: diener Date: 2011/08/24 11:34 AM Modified: /examples/kerr/figs/ qlm_spin.gp, qlm_spin.pdf Log: Modification to the spin plot. File Changes: Directory: /examples/kerr/figs/ =============================== File [modified]: qlm_spin.gp Delta lines: +8 -4 =================================================================== --- examples/kerr/figs/qlm_spin.gp 2011-08-23 23:28:55 UTC (rev 123) +++ examples/kerr/figs/qlm_spin.gp 2011-08-24 16:34:33 UTC (rev 124) @@ -3,13 +3,17 @@ set size 0.6,0.6 +fac = (16.0/12.0)**2 set xlabel 'Time (M)' -set ylabel 'Spin (M^2)' +set ylabel '{/Symbol D}S (M^2)' -plot [] [0.7:0.7004] '../by-a0.7/qlm_scalars.dat' u 1:3 w l lw 2 title 'low' -rep '../by-a0.7-med/qlm_scalars.dat' u 1:3 w l lw 2 title 'med' -rep '../by-a0.7-hi/qlm_scalars.dat' u 1:3 w l lw 2 title 'high' +print fac +plot [] [*:0.0004] '../by-a0.7/qlm_scalars.dat' u 1:($3-0.7) w l lw 2 title '{/Symbol D}S_{low}' +rep '../by-a0.7-med/qlm_scalars.dat' u 1:($3-0.7) w l lw 2 title '{/Symbol D}S_{med}' +rep '../by-a0.7-hi/qlm_scalars.dat' u 1:($3-0.7) w l lw 2 title '{/Symbol D}S_{high}' +rep '../by-a0.7-hi/qlm_scalars.dat' u 1:(fac*($3-0.7)) w l lw 2 title '1.78*{/Symbol D}S_{high}' + set term postscript enhanced color dashed set output 'qlm_spin.eps' repl File [modified]: qlm_spin.pdf Delta lines: +44 -41 =================================================================== --- examples/kerr/figs/qlm_spin.pdf 2011-08-23 23:28:55 UTC (rev 123) +++ examples/kerr/figs/qlm_spin.pdf 2011-08-24 16:34:33 UTC (rev 124) @@ -3,36 +3,27 @@ 5 0 obj <</Length 6 0 R/Filter /FlateDecode>> stream -x;³%Çqýû+® 8ê®~»PÐ¢Ar=RP`%>þ}VVeö9\"p06qv¦+»»º¦¾ýë{yÔ÷ÿøç>¿ýÛïÖû«åýoñWï¿ûõMÿÿZFyÿL]k{ÿ)u/Ûô{/õøï¿çûáí¯ -ÿñ?}~ÿ÷Oe¿×þþé¿üÂúþ-¢Û¹ÞW{XßþWß¾q_ëWþûí?>½ýöÍZ×ChzÔôðóC·ñ¨ë½Û£V7t)e<ÃRoøÐð©?¾ôÇòðç±Ï| -_oô:ÏTôÔ=4£S4z]ýa_|}N¾¶c7|hVøÔßÛcmòö^üEÍèÐúÃÑk{´¯MÞ_»)zÈÉØ)?zÛãÌs¾2óvC÷µ:ddè 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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/examples/kerr/ (Rev. 123)
by diener＠cct.lsu.edu 23 Aug '11

23 Aug '11

User: diener Date: 2011/08/23 06:28 PM Added: /examples/kerr/by-a0.7-hi/ qlm_scalars.dat /examples/kerr/by-a0.7-med/ qlm_scalars.dat /examples/kerr/by-a0.7/ qlm_scalars.dat /examples/kerr/figs/ qlm_mass.gp, qlm_mass.pdf, qlm_spin.gp, qlm_spin.pdf Log: Add data and plots for QuasiLocalMeasures calculation of mass and spin. File Changes: Directory: /examples/kerr/by-a0.7/ ================================== File [added]: qlm_scalars.dat Delta lines: <Unavailable> Directory: /examples/kerr/by-a0.7-hi/ ===================================== File [added]: qlm_scalars.dat Delta lines: <Unavailable> Directory: /examples/kerr/by-a0.7-med/ ====================================== File [added]: qlm_scalars.dat Delta lines: <Unavailable> Directory: /examples/kerr/figs/ =============================== File [added]: qlm_mass.gp Delta lines: <Unavailable> File [added]: qlm_mass.pdf Delta lines: <Unavailable> File [added]: qlm_spin.gp Delta lines: <Unavailable> File [added]: qlm_spin.pdf Delta lines: <Unavailable>

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 122)
by diener＠cct.lsu.edu 23 Aug '11

23 Aug '11

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,

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/examples/kerr/ (Rev. 121)
by diener＠cct.lsu.edu 23 Aug '11

23 Aug '11

User: diener Date: 2011/08/23 02:30 PM Added: /examples/kerr/by-a0.7-hi/ BH_diagnostics.ah1.gp /examples/kerr/by-a0.7-med/ BH_diagnostics.ah1.gp /examples/kerr/by-a0.7/ BH_diagnostics.ah1.gp /examples/kerr/figs/ ah_mass.gp, ah_mass.pdf, ah_mass_conv.gp, ah_mass_conv.pdf Log: Add Irreducible mass data and plots. File Changes: Directory: /examples/kerr/by-a0.7/ ================================== File [added]: BH_diagnostics.ah1.gp Delta lines: <Unavailable> Directory: /examples/kerr/by-a0.7-hi/ ===================================== File [added]: BH_diagnostics.ah1.gp Delta lines: <Unavailable> Directory: /examples/kerr/by-a0.7-med/ ====================================== File [added]: BH_diagnostics.ah1.gp Delta lines: <Unavailable> Directory: /examples/kerr/figs/ =============================== File [added]: ah_mass.gp Delta lines: <Unavailable> File [added]: ah_mass.pdf Delta lines: <Unavailable> File [added]: ah_mass_conv.gp Delta lines: <Unavailable> File [added]: ah_mass_conv.pdf Delta lines: <Unavailable>

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 120)
by diener＠cct.lsu.edu 23 Aug '11

23 Aug '11

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,

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/examples/kerr/ (Rev. 119)
by diener＠cct.lsu.edu 23 Aug '11

23 Aug '11

User: diener Date: 2011/08/23 11:16 AM Added: /examples/kerr/by-a0.7-hi/ mp_psi4_l4_m0_r30.00.asc /examples/kerr/by-a0.7-med/ mp_psi4_l4_m0_r30.00.asc /examples/kerr/by-a0.7/ mp_psi4_l4_m0_r30.00.asc /examples/kerr/figs/ waves_l4.gp, waves_l4.pdf, waves_l4_conv.gp, waves_l4_conv.pdf Log: Data and plots for l=4, m=0 mode. File Changes: Directory: /examples/kerr/by-a0.7/ ================================== File [added]: mp_psi4_l4_m0_r30.00.asc Delta lines: <Unavailable> Directory: /examples/kerr/by-a0.7-hi/ ===================================== File [added]: mp_psi4_l4_m0_r30.00.asc Delta lines: <Unavailable> Directory: /examples/kerr/by-a0.7-med/ ====================================== File [added]: mp_psi4_l4_m0_r30.00.asc Delta lines: <Unavailable> Directory: /examples/kerr/figs/ =============================== File [added]: waves_l4.gp Delta lines: <Unavailable> File [added]: waves_l4.pdf Delta lines: <Unavailable> File [added]: waves_l4_conv.gp Delta lines: <Unavailable> File [added]: waves_l4_conv.pdf Delta lines: <Unavailable>

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[svn:einsteintoolkit] www/about/members/ (Rev. 605)
by knarf＠cct.lsu.edu 23 Aug '11

23 Aug '11

User: knarf Date: 2011/08/23 09:49 AM Modified: /about/members/ members.txt Log: a new member - year File Changes: Directory: /about/members/ ========================== File [modified]: members.txt Delta lines: +3 -0 =================================================================== --- about/members/members.txt 2011-08-23 14:47:48 UTC (rev 604) +++ about/members/members.txt 2011-08-23 14:49:36 UTC (rev 605) @@ -94,6 +94,9 @@ Universidad Nacional AutÃ³noma de MÃ©xico Jose Manuel Torres +University of Oklahoma + Brian Friesen + University of Houston Clear Lake David Garrison

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[svn:einsteintoolkit] www/documentation/ (Rev. 604)
by knarf＠cct.lsu.edu 23 Aug '11

23 Aug '11

User: knarf Date: 2011/08/23 09:47 AM Modified: /documentation/ request_email Log: improve text slitely File Changes: Directory: /documentation/ ========================== File [modified]: request_email Delta lines: +2 -2 =================================================================== --- documentation/request_email 2011-08-19 16:49:24 UTC (rev 603) +++ documentation/request_email 2011-08-23 14:47:48 UTC (rev 604) @@ -30,6 +30,6 @@ If you continue to use the Einstein Toolkit, please join our consortium by signing up at <http://einsteintoolkit.org/about/members/join>. -Please also write to <users(a)einsteintoolkit.org> if you have problems -or questions about the examples. +Please also don't hesitate to write to <users(a)einsteintoolkit.org> if +you have problems or questions about the examples.

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 118)
by bcmsma＠astro.rit.edu 22 Aug '11

22 Aug '11

User: bmundim Date: 2011/08/22 06:38 PM Modified: / ET.tex Log: Several typos fixed as I went through the paper. Small text adjustments in the BHB section. Introduce empty objects {} to make the mixed index tensors look better. File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +37 -40 =================================================================== --- ET.tex 2011-08-22 13:13:34 UTC (rev 117) +++ ET.tex 2011-08-22 23:38:05 UTC (rev 118) @@ -1346,7 +1346,7 @@ referred to as fluxes. By calculating fluxes at cell faces, we may obtain a consistent description of the intercell values using reconstruction techniques that account for the fact that hydrodynamic -variables are not smooth and may not be finite differences accurately. +variables are not smooth and may not be finite differenced accurately. All other source terms in the evolution equations may contain only the hydrodynamic variables themselves as well as the metric variables and derivatives of the latter, since they must formally be smooth and thus @@ -1648,7 +1648,7 @@ piecewise polytropic with a thermal component designed for the application in simple models of stellar collapse. At densities below nuclear, a polytropic EOS with $\Gamma = \Gamma_1 \approx 4/3$ is -used. To mimick the stiffening of the nuclear EOS at nuclear density, +used. To mimic the stiffening of the nuclear EOS at nuclear density, the low-density polytrope is fitted to a second polytrope with $\Gamma = \Gamma_2 \gtrsim 2$. To allow for thermal contributions to the pressure due to shock heating, a gamma-law with $\Gamma = \Gamma_\mathrm{th}$ @@ -1662,7 +1662,7 @@ & & + (\Gamma_{\rm th} - 1) \rho \epsilon\,. \label{eq:hybrid_eos} \end{eqnarray}% -mIn this, $\epsilon$ denotes the total specific internal energy which +In this, $\epsilon$ denotes the total specific internal energy which consists of a polytropic and a thermal contribution. In iron core collapse, the pressure below nuclear density is dominated by the pressure of relativistically degenerate electrons. For this, one sets @@ -1784,7 +1784,7 @@ \frac{\beta}{\alpha} L^2\right)^{-1} \rho \Theta \end{equation} where $\rho$ is a strictly positive weight, $L^2$ is the Laplacian -of the 2D metric(?), and $\alpha$,$\beta$, and $\ell_\mathrm{max}$ are +of the 2D metric(?), and $\alpha$, $\beta$, and $\ell_\mathrm{max}$ are free parameters. Decomposing $h(\theta,\phi)$ onto a basis of spherical harmonics, the coefficients $a_{\ell m}$ evolve iteratively towards a solution as @@ -1800,7 +1800,7 @@ The \codename{AHFinderDirect} module~\cite{Thornburg:2003sf} is a faster alternative to \codename{AHFinder}. Its approach is to view Eq.\ref{eq:ah_theta} as an elliptic PDE for $h(\theta,\phi)$ on $S^2$ -using standard finite differencing methods. Rewriting Eq.~\ref{ah_theta} +using standard finite differencing methods. Rewriting Eq.~\ref{eq:ah_theta} in the form \begin{equation} \Theta \equiv \Theta\left(h,\partial_u h,\partial_{uv}h; @@ -1825,8 +1825,8 @@ M &=& \frac{1}{16\pi} \int_\Omega d^3 x \left[ e^{5 \phi} \left( 16 \pi \rho + \tilde{A}_{ij} \tilde{A}^{ij} - \frac23 K^2 \right) - e^\phi \tilde{R} \right] \\ - J_i &=& \frac{1}{8 \pi} \epsilon_{ij}^k \int_\Omega d^3 x \left[ - e^{6\phi}\left( \tilde{A}^j_k + \frac23 x^j \tilde{D}_k K + J_i &=& \frac{1}{8 \pi} \epsilon_{ij}{}^k \int_\Omega d^3 x \left[ + e^{6\phi}\left( \tilde{A}^j{}_k + \frac23 x^j \tilde{D}_k K - \frac12 x^j \tilde{A}_{\ell n} \partial_k \tilde{\gamma}^{\ell n} + 8 \pi x^j S_k \right) \right] \end{eqnarray} @@ -1837,7 +1837,7 @@ As the surface terms required when converting a surface integral to a volume integral are neglected, this procedures hold under the assumption that the integrals of the integrands $\tilde{D}^i e^\phi$ and -$e^{6\phi} \epsilon_{ij}^k x^j \tilde{A}^\ell_k$ on the boundaries of +$e^{6\phi} \epsilon_{ij}{}^k x^j \tilde{A}^\ell{}_k$ on the boundaries of the computational domain vanish. The ADM mass and angular momentum can also be calculated by the module \codename{Extract}, from the variables stored in the base modules, as surface integrals~\cite{Bowen:1980yu} @@ -1864,9 +1864,9 @@ this sense, \codename{QuasiLocalMeasures} can provide gauge-independent measures of e.g.\ mass and angular momentum of regions of the spacetime enclosed in a two-sphere. -\codename{QuasiLocalMeasures} takes as input a horizon surface any +\codename{QuasiLocalMeasures} takes as input a horizon surface, or any other surface that the user specifies, such as a large coordinate -sphere, and calculates can calculate mass, angular momentum, or other +sphere, and calculates mass, angular momentum, or other quantities such as the Weyl or Ricci scalars, multipole moments of the horizon geometry, or the three-volume element of the horizon world tube. @@ -1984,8 +1984,8 @@ base modules described in Sec.~\ref{sec:base_modules}, explicitly written as: \begin{eqnarray} - H &=& R - K^i_j K^j_i + K^2 - 16 \pi \rho \\ - M_i &=& \nabla_j K_i^j - \nabla_i K - 8 \pi S_i + H &=& R - K^i{}_j K^j{}_i + K^2 - 16 \pi \rho \\ + M_i &=& \nabla_j K_i{}^j - \nabla_i K - 8 \pi S_i \end{eqnarray} where $S_i=-\frac{1}{\alpha} \left( T_{i0} - \beta^j T_{ij} \right)$. The difference between these modules lies in how they access the stress @@ -2015,7 +2015,7 @@ The grid functions provided by \codename{TmunuBase} are: \begin{itemize} - \item The time compenent $T_{00}$ {\tt eTtt} + \item The time component $T_{00}$ {\tt eTtt} \item The mixed components $T_{0i}$ {\tt eTtx}, {\tt eTty}, {\tt eTtz} \item The spatial components $T_{ij}$ {\tt eTxx}, {\tt eTxy}, {\tt eTxz}, {\tt eTyy}, {\tt eTyz}, {\tt eTzz} \end{itemize} @@ -2072,8 +2072,8 @@ the numerical resolution do not affect the extent of the physical domain, i.e.\ that the discrete domains converge to the physical domain in the limint of infinite resolution. -The Einstein Toolkit provided thorn \codename{CoordBase} facilates -this specification of the simulation domain independend of the actual +The Einstein Toolkit provides thorn \codename{CoordBase} that facilitates +this specification of the simulation domain independent of the actual evolution thorn used. The domain is specified at run time via a parameter file at the same time as parameters describing the physical system are specified. @@ -2085,9 +2085,8 @@ grid when creating the hierarchy of grids, automatically ensuring a consistent grid description between the two thorns. Evolution thorns such as \codename{McLachlan} use the domain -information information to decide which points are evolved and therefore require the -evaluation of the righ-hand-side expression and which ones are set via +evaluation of the right-hand-side expression and which ones are set via boundary or symmetry conditions. \subsubsection{Symmetries, Boundary Conditions.} @@ -2169,12 +2168,12 @@ tracking objects as they move through the domain. One can also add or remove stacks if e.g.\ the number of objects changes. Fully AMR based on a local error estimate is supported by \codename{Carpet} however the -Einstein Toolkit does not presently provide a suitable redgridding thorn +Einstein Toolkit does not presently provide a suitable regridding thorn to create such a grid. If initial conditions are constructed outside of Carpet (which is often the case), then the initial mesh hierarchy has to be defined manually. -In order to fascilate the description of the mesh hierarchy the\ +In order to facilitate the description of the mesh hierarchy the\ Einstein toolkit provides two regridding modules in the \codename{CarpetRegrid} and \codename{CarpetRegrid2} thorns. Both thorns primarily support box-in-box type refined meshes, which is @@ -2263,7 +2262,7 @@ \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 resultion run. The extraction was + 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 shows for the real part of the $\ell =2, m=0$ waveforms the @@ -2301,7 +2300,8 @@ points. Along with the Bowen-York analytic solution for the momentum constraints, the spectral solution is then interpolated in the simulation domain, representing thus a constrained -initial data $\{\gamma_{ij},K_{ij}\}$ +initial data $\{\gamma_{ij},K_{ij}\}$. Its evolution is performed +by the \codename{McLachlan} module. \begin{table}[!ht] \caption{Initial data parameters for a non-spinning equal mass @@ -2333,8 +2333,8 @@ decomposition takes place. Figure~\ref{fig:tracks_waveform} shows the two punctures tracks -throughout all phases of the binary dynamics. There tracks are -provided by \codename{PunctureTracker} module. Also in the same +throughout all phases of the binary dynamics. Tracks are +provided there by the \codename{PunctureTracker} module. Also in the same plot we have recorded every $10M$ of evolution, the intersection of the apparent horizon $2$-surface with the $z=0$ plane. A common horizon is first observed at $t=116M$. These apparent @@ -2357,15 +2357,15 @@ The number of grid points on the sphere was set to be $[n_{\theta},n_{\phi}]=[120,240]$, what gives an angular resolution in this case of the order of $2.6 \times 10^{-2}$ -radians. +radians, +and an error of the same order, since the surface integrals were +calculated by midpoint rule -- a first order accurate method. %BCM: This is certainly not correct. Give a better % thought later: -and an error of the same order, since the surface integrals were -calculated by midpoint rule -- a first order accurate -method. Note however that this error is still -%at least an order of magnitude -smaller than, for example, the metric truncation error -at this extraction region ($\sim 6.25 \times 10^{-2}$). +%Note however that this error is still +%%at least an order of magnitude +%smaller than, for example, the metric truncation error +%at this extraction region ($\sim 6.25 \times 10^{-2}$). In order to evaluate the convergence of the numerical solution, we ran three different simulations with different @@ -2414,20 +2414,18 @@ In Figure~\ref{fig:amp_phs_convergence}, we show the convergence of the amplitude and phase of the Weyl scalar by plotting the logarithm of the absolute value of the differences between two levels -of resolution. The differences clearly converges to zero as the resolution +of resolution. The differences clearly converge to zero as the resolution is increased. We also show that the most appropriate convergence factor seems to be the one corresponding to an $8$-th order accurate finite difference approximation. \begin{figure} -% \begin{center} \includegraphics[width=0.45\textwidth]{examples/bbh/figs/tracks} \includegraphics[width=0.45\textwidth]{examples/bbh/figs/mp_psi4_l2_m2_r60} -% \end{center} \caption{On the left panel, we have plotted the tracks corresponding to the evolution of two punctures initially located on the $x$-axis at $x=\pm 3$. The solid blue line represents puncture 1, while the dashed red line the -puncture two. The circular dotted green lines are the intersections of the +puncture 2. The circular dotted green lines are the intersections of the apparent horizons with the $z=0$ plane plotted every $10M$ of binary evolution. The common horizon arises at $t=116M$. On the right panel, we have plotted the real (solid blue line) and imaginary (dotted red line) @@ -2437,12 +2435,11 @@ \end{figure} \begin{figure} -% \begin{center} \includegraphics[width=0.45\textwidth]{examples/bbh/figs/amp_convergence_memehhi_8th} \includegraphics[width=0.45\textwidth]{examples/bbh/figs/phase_convergence_memehhi_8th} -% \end{center} \caption{Weyl scalar amplitude (left panel) and phase (right panel) -convergence: in progress... } +convergence. The dotted vertical green line indicates the evolution time +in which the Weyl scalar frequency reaches $\omega=0.2/M$. in progress... } \label{fig:amp_phs_convergence} \end{figure} @@ -2451,7 +2448,7 @@ In figure~\ref{fig:tov_mode_spectrum} we show the power spectrum of the central density oscillations computed from a full 3D relativistic hydrodynamics -simulation using GRHydro compared to the cooresponding frequencies as obtained +simulation using GRHydro compared to the coresponding frequencies as obtained with perturbative techniques\todo{thanks}. Clearly the 3D simulation can correctly represent the fundamental mode frequency and its overtones of the neutron star. @@ -2525,7 +2522,7 @@ One of the desirable additions of physics is a proper treatment of radiation, in particular neutrinos. Another desirable addition would be some -approximation of emmission of electro-magnetic waves. Radiation transport is, +approximation of emission of electro-magnetic waves. Radiation transport is, even compared to the full GRMHD problem, computationally very expensive, especially in three dimensions. Several members of the CIGR team are involved in the NSF project PetaCactus, which aims to explore these possibilities. @@ -2548,7 +2545,7 @@ Yet another important goal is to increase the scalability of the Carpet AMR infrastructure. It has been shown that good scaling is limited to less than -a few thousend proccesses, for some of the most used simulation scenarios. +a few thousand processes, for some of the most used simulation scenarios. In this case, work is already in progress to eliminate this bottle-neck. On the other hand, a production simulation is typically composed from a large number of components, and all of them have to scale well to achieve overall

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/examples/tov/ (Rev. 117)
by knarf＠cct.lsu.edu 22 Aug '11

22 Aug '11

User: knarf Date: 2011/08/22 08:13 AM Modified: /examples/tov/ mode_spectrum.pdf, mode_spectrum.svg /examples/tov/tov_2/ hydrobase::rho.maximum.asc.bz2, hydrobase::rho.maximum.asc.spectrum.bz2 Log: update figure File Changes: Directory: /examples/tov/ ========================= File [modified]: mode_spectrum.pdf Delta lines: +0 -0 =================================================================== (Binary files differ) File [modified]: mode_spectrum.svg Delta lines: +73 -42 =================================================================== --- examples/tov/mode_spectrum.svg 2011-08-22 13:12:23 UTC (rev 116) +++ examples/tov/mode_spectrum.svg 2011-08-22 13:13:34 UTC (rev 117) @@ -93,48 +93,79 @@ </g> </g> <g style="fill:none; color:red; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter"> - <path d='M37.5,20.9 L40.3,75.4 L55.0,94.0 L69.7,103.6 L84.5,98.6 L99.2,73.6 L113.9,89.2 L128.6,124.2 - L143.3,141.2 L158.0,161.8 L172.8,165.2 L187.5,170.0 L202.2,161.8 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