[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 51)

[bcmsma at astro.rit.edu]

User: bmundim
Date: 2011/04/11 09:54 AM

Modified:
 /
  ET.tex

Log:
 An update to initial data section (by Josh/Bruno).

File Changes:

Directory: /
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File [modified]: ET.tex
Delta lines: +80 -34
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--- ET.tex	2011-04-11 08:32:32 UTC (rev 50)
+++ ET.tex	2011-04-11 14:54:30 UTC (rev 51)
@@ -659,63 +659,109 @@
 general relativistic simulations, including both vacuum and hydrodynamical
 configurations.  
 These include modules used primarily for testing of various components, as
-well as physically motivated configurations that e.g.\ describe single or
-binary black holes and/or neutron stars.  Many of the modules
+well as physically motivated configurations that 
+describe, for example, single or binary black holes and/or neutron stars.  
+Many of the modules
 are self-contained, consisting of either all the code to generate exact 
 initial solutions or the numerical tools required to construct solutions 
 known semi-analytically. Others, though, require the installation of other 
 numerical software packages that are included in the toolkit as external 
-libraries. One example is the {\tt TwoPunctures}
+libraries. One example is the \codename{TwoPunctures}
 module~\cite{Ansorg:2004ds}, commonly 
 used in numerical relativity to generate binary black hole data, which makes
 use of the GNU Scientific Library [GSL;~\cite{GSL:web,Galassi:2009}].
 Several modules have also been implemented to read in datafiles generated by
-the {\tt Lorene code}~\cite{Lorene:web,Gourgoulhon:2000nn}.
+the {\tt Lorene code}~\cite{Lorene:web,Gourgoulhon:2000nn}. 
 
-Initial data setup is in most cases clearly separated by the following
+Initial data setup is in most cases clearly separated from the following
 evolution. Typically, initial data routines provide the data in form of the
 quantities defined in the Base modules (see section~\ref{sec_base_modules}),
-while the evolution modules may convert these quantities to quantities
+and the evolution modules will convert these quantities to quantities
 used for the evolution later. For example, an initial data module must supply 
 $g_{ij}$, the spatial 3-metric, and $K_{ij}$, the extrinsic curvature.  
-The evolution scheme however, typically makes use of the BSSN formalism.
 The conversion between the physical and conformal metric and extrinsic 
-curvature is handled solely within evolution modules. Optionally, many
-initial data modules also supply values 
+curvature is handled solely within evolution modules, which are responsible for calculating the conformally related three metric $\tilde{\gamma}_{ij}$, the conformal factor $\psi$, the conformal traceless extrinsic curvature $\tilde{A}_{ij}$, and the trace of the extrinsic curvature $K$, as well as initializing the BSSN variable $\tilde{\Gamma}^i$ should that be the evolution formalism chosen. 
+Optionally, many initial data modules also supply values 
 for the lapse and shift vector, and in some cases time derivatives as well.
-The initial data routines currently implemented include the following modules.
+It is important to note, though, that many dynamical calculations typically run better from initial gauge choices set by ansatz rather than those derived from stationary approximations that are incompatible with the gauge evolution equations.  In particular, conformal thin-sandwich initial data for binaries include solutions for the lapse and shift that are frequently replaced by simpler analytical values that lead to more circular orbits under standard `moving puncture' gauge conditions (see, e.g., \cite{Etienne:???} and other works).
 
+\subsubsection{Simple Vacuum initial data}
+
 Vacuum spacetime tests can be provided by \codename{IDConstraintViolate}, which
 provides a vacuum spacetime explicitly violating the constraint equations, and
-\codename{Exact}, a set of exact, and possibly Lorentz-boosted spacetimes in
-various coordinates.
+\codename{Exact}, a set of exact spacetimes in
+various coordinates, including Lorentz-boosted versions of traditional solutions..
 Vacuum gravitational wave configurations can be obtained by using either
-\codename{IDBrillData}, providinga Brill wave spacetime~\cite{Brill:1959zz},
+\codename{IDBrillData}, providing a Brill wave spacetime~\cite{Brill:1959zz},
 or \codename{IDLinearWaves}, for a spacetime containing a linear gravitational
 wave.
-Possible black hole configurations include \codename{IDAnalyticBH}, generating
-various analyticly known black hole configurations, as well as
+Single black hole configurations include \codename{IDAnalyticBH}, which generates
+various analytically known black hole configurations, as well as
 \codename{IDAxibrillBH}, \codename{IDAxiOddBrillBH}, \codename{DistortedBHIVP}
-and \codename{RotatingDBHIVP}, generating single black hole systems, distorted in
-various different ways. \codename{Meudon\_Bin\_BH} can read in binary black hole
-initial data computed by the Lorene~\cite{TODO} code.
+and \codename{RotatingDBHIVP}, which perturbed isolated black holes.
+
+\subsubsection{TwoPunctures: Binary Black Holes and extensions}
+A substantial fraction of the published work on the components of the Einstein toolkit involves the evolution of binary black hole systems.
+The most widely used routine to generate initial data for these is the \codename{TwoPunctures} code, described originally in \cite{Ansorg:}, which solves the binary puncture equations for a pair of black holes \cite{Brandt:???}.
+To do so, one assumes the extrinsic curvature for each black hole corresponds to the Bowen-York form \cite{Bowen:???}, 
+\begin{eqnarray*}
+K_{(n)}^{ij}&=&\frac{3}{2r^2}(P^in^j+P^jn^i-(\gamma^{ij}-n^in^jP^kn_k))\\
+&&+\frac{3}{r^3}(\varepsilon^{ikl}S_kn_ln^j+\varepsilon^{jkl}S_kn_ln^i)
+\end{eqnarray*}
+where the subscript $(n)$ refers to the contribution from black hole $n=1,2$, the 3-momentum is $P^i$, the BH spin angular momentum is $S_i$, the conformal 3-metric $\gamma^{ij}$ is assumed to be flat, i.e., $\gamma_{ij}=\eta_{ij}$, and $n^i=x^i/r$ is the Cartesian normal vector relative to the position of each BH in turn.  This solution automatically satisfies the momentum constraint, and the Hamiltonian constraint may be written as an elliptical equation for the conformal factor ($g_{ij}=\psi^4\gamma_{ij}$):
+\begin{eqnarray*}
+\Delta \psi+\frac{1}{8}K^{ij}K_{ij}\psi^{-7}=0
+\end{eqnarray*}
+Decomposing the conformal factor into a singular analytical term and a regular term $u$, such that
+\begin{eqnarray*}
+\psi = \frac{m_1}{2r_1}+\frac{m_2}{2r_2}_+u\equiv \frac{1}{\alpha}+u
+\end{eqnarray*}
+where $m_i$ and $r_i$ are the mass of and distance to each BH, respectively, the Hamiltonian constraint may be written
+\begin{eqnarray*}
+\Delta u +\left[\frac{1}{8}\alpha^7K^{ij}K_{ij}\right](1+\alpha u)^{-7}
+\end{eqnarray*}
+subject to the boundary condition $u\rightarrow 1$ as $r\rightarrow\infty$.  In Cartesian coordinates, the function $u$ is infinitely differentiable everywhere except the two puncture locations.  \codename{TwoPunctures} resolves this problem by performing a coordinate transformation modeled on confocal elliptical/hyperbolic coordinates, which transforms the spatial domain into a finite cube with the puncture locations mapped to two parallel edges, as can be seen in Fig.~\ref{fig:TP_BHNS_coordinates}.  Since $u$ is smooth everywhere in the interior of the remapped domain, expansions into modes in these coordinates are {\em spectrally convergent}, and thus capable of extremely high accuracy.  In practice, the field is expanded into Chebyshev modes in the quasi-elliptical and quasi-hyperbolic coordinates, and in Fourier modes around the axis connecting the two BHs.  The elliptic solver uses a stabilized biconjugate gradient method to achieve rapid solutions.
+
+\begin{figure}
+ \label{fig:TP_BHNS_coordinates}
+ \centering\includegraphics[width=0.5\textwidth]{TwoPunctures_grid_BHNS}\\
+ \caption{Example of a TwoPunctures coordinate system for BH-NS binary initial data}
+\end{figure}
+
+ \codename{Meudon\_Bin\_BH} can read in binary black hole
+initial data computed by the Lorene~\cite{Lorene:web,Gourgoulhon:2000nn} code.
 Finally, the most used type of black hole initial data is provided by the module
 \codename{TwoPunctures}, which generates accurate binary black-hole initial data.
 
 Initial data to test different parts of hydrodynamics evolution systems can
-be provided by \codename{GRHydro\_InitData}, e.g.\ for shock tests,
+be provided by \codename{GRHydro\_InitData}, for example, initial data for shock tests,
 \codename{Hydro\_InitExcision} for tests involving hydro-excision,
 \codename{TOVSolver} for Tolman-Oppenheimer-Volkov type neutron star initial data,
 \codename{Meudon\_Bin\_NS} and \codename{Meudon\_Mag\_NS} read in binary and
 magnetized neutron star data computed using Lorene, respectively. Finally,
 the \codename{TwoPunctures} module can also be used to construct neutron star
 black hole binary initial data, when being coupled with \codename{TOVSolver}.
-\begin{figure}
- \label{fig:TP_BHNS_coordinates}
- \centering\includegraphics[width=0.5\textwidth]{TwoPunctures_grid_BHNS}\\
- \caption{Example of a TwoPunctures coordinate system for BH-NS binary initial data}
-\end{figure}
 
+We turn our attention next to a brief discussion of the capabilities of each
+of the modules mentioned earlier, as well as their particular implementation.
+
+%\subsubsection{IDConstraintViolate}
+%\subsubsection{Exact}
+%\subsubsection{IDBrillData}
+%\subsubsection{IDLinearWaves}
+%\subsubsection{IDAnalyticBH}
+%\subsubsection{IDAxibrillBH}
+%\subsubsection{IDAxiOddBrillBH}
+%\subsubsection{DistortedBHIVP}
+%\subsubsection{RotatingDBHIVP}
+%\subsubsection{Meudon\_Bin\_BH}
+%\subsubsection{Meudon\_Bin\_NS}
+%\subsubsection{Meudon\_Mag\_NS}
+%\subsubsection{GRHydro\_InitData}
+%\subsubsection{Hydro\_InitExcision}
+%\subsubsection{TOVSolver}
+%\subsubsection{TwoPunctures}
+
 \subsection{Equation of States}\pages{1 Christian}
 
 \subsection{Spacetime Curvature and Hydrodynamics Evolution}
@@ -1468,16 +1514,16 @@
 of spacetime~\ref{eq:kasner}, along with the associated error for a sequence of 
 time resolutions.
 
-\begin{figure}
-\includegraphics[width=0.45\textwidth]{kasner.png}
-\includegraphics[width=0.45\textwidth]{err.png}
-\caption{Left: the evolution of a vacuum spacetime of the type~\ref{eq:gowdyT3},
-with $P=Q=\lambda=0$; the initial data are chosen as
-$\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$.
-Right: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$;
-the errors are scaled according to the expectation for fourth-order convergence.
-\label{fig:kasner}}
-\end{figure}
+%\begin{figure}
+%\includegraphics[width=0.45\textwidth]{kasner.png}
+%\includegraphics[width=0.45\textwidth]{err.png}
+%\caption{Left: the evolution of a vacuum spacetime of the type~\ref{eq:gowdyT3},
+%with $P=Q=\lambda=0$; the initial data are chosen as
+%$\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$.
+%Right: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$;
+%the errors are scaled according to the expectation for fourth-order convergence.
+%\label{fig:kasner}}
+%\end{figure}
 
 
 \section{Future Work\pages{1 Frank}}