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