User: jfaber Date: 2011/11/08 10:29 PM

Modified: / ET.tex

Log: A few more edits to the Examples section. I'm pretty much done for now, the paper looks very good to me.

File Changes:

Directory: / ============

File [modified]: ET.tex Delta lines: +31 -32 =================================================================== --- ET.tex 2011-11-08 17:31:14 UTC (rev 178) +++ ET.tex 2011-11-09 04:29:46 UTC (rev 179) @@ -2645,10 +2645,10 @@ The examples in the previous subsections did not include the evolution of matter within a relativistic spacetime. One interesting test of a coupled matter-spacetime evolution is to measure the eigenfrequencies of a stable TOV -star~\todo{cite papers also doing this}. These eigenfrequencies can be compared +star~(see, e.g.,~\cite{Gourgoulhon:1991aa,Romero:1996aa,Shibata:1998sg,Font:2001ew,Shibata:2003iy}). \todo{Do we need more refs? Five is decent for an e.g.} These eigenfrequencies can be compared to values known from linear perturbation theory.

-This test uses an equilibrium configuration of a self-gravitating fluid +We begin our simulations with a self-gravitating fluid sphere, described by a polytropic equation of state. This one-dimensional solution is obtained by the code described in section~\ref{sec:TOVSolver}, and is interpolated on the three-dimensional, computational evolution grid. @@ -2656,14 +2656,14 @@ \codename{McLachlan} and the hydrodynamics evolution system implemented in \codename{GRHydro}.

-Specifically for this test, we set up a stable TOV star described by a +For the test described here, we set up a stable TOV star described by a polytropic equation of state $p=K\rho^\Gamma$ with $K=100$ and $\Gamma=2$, and an initial central density of $\rho_c=1.28\times10^{-3}$. This model can be taken to represent a non-rotating NS with a mass of $M=1.4\mathrm{M}_\odot$. The computational domain is a cube of length $640\mathrm{M}$ with a base resolution of $2\mathrm{M}$ ($4\mathrm{M}$, $8\mathrm{M}$) in each dimension. Four additional grids refine the region -around the star located at the origin, each doubling the resolution, with sizes +around the star centered at the origin, each doubling the resolution, with sizes of $120\mathrm{M}$, $60\mathrm{M}$, $30\mathrm{M}$ and $15\mathrm{M}$, resulting in a resolution of $0.125\mathrm{M}$ ($0.25\mathrm{M}$, $0.5\mathrm{M}$) across the entire star. @@ -2691,13 +2691,13 @@ hydrodynamics simulation, compared to the corresponding frequencies as obtained with perturbative techniques, kindly provided to use by Kentaro Takami and computed using the method described in~\cite{Yoshida:1999vj}. The PSD was computed -using the entire time series of the high-resolution run, removing the linear -trend, averaging over Hanning windows overlapping half the signal length after -padding the signal to five time it's length. The agreement of the frequencies -of the fundamental mode and the first three overtones is clearly visible, but -we are not able to show the same for higher overtones, at this resolution. -We expect to be able to resolve higher overtones when using even higher -resolution, but did not pursue further because of the involved computational +using the entire time series of the high-resolution run, by removing the linear +trend and averaging over Hanning windows overlapping half the signal length after +padding the signal to five time its length. The agreement of the + fundamental mode and first three overtone frequencies is clearly visible, but +are limited beyond this by the finite numerical resolution. +Higher overtones should be measurable with higher +resolution, but at substantial computational cost.

\begin{figure} @@ -2718,7 +2718,7 @@ code lies between $1$ and $2$. It cannot be higher than $2$ due to the directional flux-split algorithm which is of second order. Depending on solution itself, the hydrodynamics code is only of first order in particular -regions, e.g., at extrema (like the center of the star), or at the star +regions, e.g., at extrema (like the center of the star), or at the stellar surface.

Figure~\ref{fig:tov_ham_conv} shows the order of convergence of the Hamiltonian @@ -2736,7 +2736,7 @@ $r=5\mathrm{M}$. The observed convergence order of about $1.5$ at the center of the star is lower then the general second order of the hydrodynamics evolution scheme. This is expected because the scheme's convergence rate drops to first - order at extrema or shocks, like the stellar center or the star surface. + order at extrema or shocks, like the stellar center or surface. Consequently, the observed convergence order about half-way between the stellar center and surface is higher than $1.5$, but most of the time below $2$.} @@ -2744,13 +2744,13 @@

\subsection{Collapse\pages{2 Christian, Roland}} \label{sec:collapse_example} -The previous examples dealt either with a preexisting black hole (BBH) or -with a smooth singularity free spacetime (TOV oscillations). The evolution +The previous examples dealt either with preexisting BHs, either single or in a binary, or +with a smooth singularity free spacetime, as in the case of the TOV star. The evolution codes in the toolkit however are also able to handle the dynamic formation of a singularity as a star collapses -into a black hole. As a simple example of this process we study the collapse -of a non-rotating TOV star into a black hole. We create initial data as in -section~\ref{sec:tov_oscilations} using $\rho_c=3.154e-3$ and $K_{ID} = 100$, +into a BH, and as a simple example of this process we study the collapse +of a non-rotating TOV star. We create initial data as in +section~\ref{sec:tov_oscillations} using $\rho_c=3.154e-3$ and $K_{ID} = 100$, $\Gamma = 2$ yielding a star of mass $1.67,M_\odot$. As is common in these situations~\cite{Baiotti:2005vi} we trigger collapse by reducing the pressure support after initial data has been constructed @@ -2779,7 +2779,7 @@ The outermost box has a radius of $R_0 = 204.8,M_\odot$ and a resolution of $3.2,M_\odot$ ($2.4,M_\odot$, $1.6,M_\odot$, $0.6,M_\odot$ for higher convergence levels). -Around the star which is centered on the origin we stack $5$ extra boxes of +Around the star, centered on the origin, we stack $5$ extra boxes of approximate size $8\times2^\ell,M_\odot$ for $0 \le \ell \le 4$, where the resolution on each finer % RH: there is a likely typo in the paramter file which creates boxes of @@ -2819,7 +2819,7 @@ approximate location of the star surface and the circumferential radius of the apparent horizon once a horizon is found in the simulation. Clearly the apparent -horizon is found at approximately the same time as the star's size approaches +horizon is first found at approximately the same time as when the star's size approaches its Schwarzschild radius. In Figure~\ref{fig:tov_collapse_H_convergence_at0} we display the convergence factor for the Hamiltonian constraint at the center of the collapsing star. @@ -2880,26 +2880,26 @@

\section{Conclusion and Future Work} In this article, we described the Einstein Toolkit, a collection -of freely available and easy to use computational codes for numerical +of freely available and easy-to-use computational codes for numerical relativity and relativistic astrophysics. The code details and example results present in this article represent the state of the Einstein Toolkit in its release ET_2011_05 ``Curie,'' released on April 21, 2011.

-What was presented here is but a snapshot of the Einstein Toolkit's -ongoing development whose ultimate goal it is to provide an +The work presented here is but a snapshot of the Einstein Toolkit's +ongoing development, whose ultimate goal it is to provide an open-source set of robust baseline codes to realistically and reproducibly model the whole spectrum of relativistic astrophysical phenomena including, but not limited to, isolated black holes and neutron stars, binary black hole coalescence in vacuum and gaseous -environs, double neutron star and neutron star -- black hole mergers, +environments, double neutron star and neutron star -- black hole mergers, core-collapse supernovae, and gamma-ray bursts.

For this, much future work towards including proper treatments of magnetic fields, more complex equations of state, nuclear reactions, neutrinos, and photons will be necessary and will need to be matched by improvements in infrastructure (e.g., more flexible AMR on general -grids) and computing hardware for the required fully coupled 3D, +grids) and computing hardware for the required fully coupled 3-D, multi-scale, multi-physics simulations to become reality.

%%% One of the desirable additions of physics is a proper treatment of @@ -2922,18 +2922,17 @@ whole community by integrating their CCE routines into the Einstein Tookit release 2011_11 ``Maxwell,'' which will be described elsewhere.

-A second example for a much needed improvement of an existing method -is to transition to cell-centered AMR for GR hydrodynamic simulations, +A second much needed improvement of our existing methods +is a transition to cell-centered AMR for GR hydrodynamic simulations, which would allow for exact flux conservation across AMR interfaces via a refluxing step that adjusts coarse and/or fine grid fluxes for consistency (e.g., \cite{Berger:1984zza}). This is also a prerequisite for the constrained transport method \cite{Toth:00} for ensuring the divergence-free condition for the magnetic field in a future -implementation of GRMHD within the Einstein Toolkit. Work towards +implementation of GRMHD within the Einstein Toolkit. Work towards cell-centered AMR, refluxing, and GRMHD is underway and will be reported in a future publication.

- %%% CDO: We have not even talked about the current MHD implementation, %%% how can we talk about its improvement? %%% @@ -2949,7 +2948,7 @@

Yet another important goal is to increase the scalability of the {\tt Carpet} AMR infrastructure. As we have shown, good scaling is -limited to only a few thousand processes for some of the most used +limited to only a few thousand processes for some of the most widely used simulation scenarios. Work is in progress to eliminate this bottle-neck. On the other hand, a production simulation is typically composed of a large number of components, and even analysis and I/O @@ -2968,8 +2967,8 @@ National Science Foundation under the grant numbers 0903973/0903782/0904015 (CIGR@). Related grants contribute directly and indirectly to the success of CIGR, including NSF OCI-0721915, NSF -OCI-0725070, NSF OCI-0905046, and NSF OCI 0941653, NSF -AST-0855535, and NASA 08-ATFP08-0093. Results presented in this article were obtained through +OCI-0725070, NSF OCI-0832606, NSF OCI-0905046, and NSF OCI-0941653, NSF +AST-0855535, NSF DMS-0820923, and NASA 08-ATFP08-0093. Results presented in this article were obtained through computations on the Louisiana Optical Network Initiative under allocation loni_cactus05 and loni_numrel06, as well as on the NSF Teragrid under allocations TG-MCA02N014 and TG-PHY060027N, DOE repository m152, HLRB at