User: bmundim
Date: 2011/04/11 05:24 PM
Modified:
/
ET.tex
Log:
Break lines around column 80 and a few edits.
File Changes:
Directory: /
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File [modified]: ET.tex
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--- ET.tex 2011-04-11 15:47:55 UTC (rev 53)
+++ ET.tex 2011-04-11 22:24:34 UTC (rev 54)
@@ -35,6 +35,8 @@
\newcommand{\comment}[1]{{\color{blue}$\blacksquare$~\textsf{[Comment: #1]}}}
\newcommand{\pages}[1]{{\color{blue}$\blacksquare$~\textsf{[#1]}}}
+\newcommand{\BCM}[1]{{\bf \color{blue} [BCM: #1] }} %BCM comments
+
% Don't use tt font for urls
\urlstyle{rm}
@@ -680,17 +682,27 @@
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 conversion between the physical and conformal metric and extrinsic
-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
+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.
-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).
+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 spacetimes in
-various coordinates, including Lorentz-boosted versions of traditional solutions..
+various coordinates, including Lorentz-boosted versions of traditional solutions.
Vacuum gravitational wave configurations can be obtained by using either
\codename{IDBrillData}, providing a Brill wave spacetime~\cite{Brill:1959zz},
or \codename{IDLinearWaves}, for a spacetime containing a linear gravitational
@@ -698,41 +710,72 @@
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}, which perturbed isolated black holes.
+and \codename{RotatingDBHIVP}, which introduce perturbations to isolated black holes.
\subsubsection{Hydrodynamics Tests}
-Initial data to test different parts of hydrodynamics evolution systems are provided by \codename{GRHydro\_InitData}. The module includes several shock tube problems that may be evolved in any of the Cartesian directions or diagonally, all of which have been widely used throughout the field to evaluate solvers \cite{Sod,Blast,etc}. Conservative to primitive variable conversion and vice versa are included, as are tests to check on the reconstruction of hydrodynamical variables at cell faces (see Sec.~\ref{GRHydro???} for more on this). Along similar lines, the \codename{Hydro\_InitExcision} module includes tests for hydrodynamics in the presence of an excised region.
+Initial data to test different parts of hydrodynamics evolution systems are provided
+by \codename{GRHydro\_InitData}. The module includes several shock tube problems
+that may be evolved in any of the Cartesian directions or diagonally, all of which
+have been widely used throughout the field to evaluate solvers \cite{Sod,Blast,etc}.
+Conservative to primitive variable conversion and vice versa are included, as are
+tests to check on the reconstruction of hydrodynamical variables at cell faces
+(see Sec.~\ref{GRHydro???} for more on this). Along similar lines, the
+\codename{Hydro\_InitExcision} module includes tests for hydrodynamics in the
+presence of an excised region.
\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:???},
+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 ($g_{ij}=\psi^4\gamma_{ij}$):
+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
+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
+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}. The coordinate transformation is
+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}.
+The coordinate transformation is
\begin{eqnarray*}
x&=&b\frac{A^2+1}{A^2-1}\frac{2B}{1+B^2}\\
y&=&b\frac{2A}{1-A^2}\frac{1-B^2}{1+B^2}\cos\phi\\
z&=&b\frac{2A}{1-A^2}\frac{1-B^2}{1+B^2}\sin\phi
\end{eqnarray*}
-which maps $\mathcal{R}^3$ onto $0\le A\le 1$ (the elliptical quasi-radial coordinate), $-1\le B\le 1$ (the hyperbolic quasi-latitudinal coordinate), and $0\le\phi<2\pi$ (the longitudinal angle).
-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-radial and quasi-latitudinal 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.
+which maps $\mathcal{R}^3$ onto $0\le A\le 1$ (the elliptical quasi-radial coordinate),
+$-1\le B\le 1$ (the hyperbolic quasi-latitudinal coordinate), and $0\le\phi<2\pi$
+(the longitudinal angle). 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-radial and quasi-latitudinal 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}
@@ -742,18 +785,43 @@
\subsubsection{Lorene-based binary data}
-The ET contains three routines that can read in publicly available data generated by the {\tt Lorene} code \cite{loreneweb,Gourgoulhon:2000nn}, though it does not currently include the capability of generating such data from scratch. For a number of reasons, such functionality is not truly required; in particular, {\tt Lorene} is a serial code and there is no time-savings at all to call it as an ET initial data generator. Also, it is not guaranteed to be convergent for an arbitrary set of parameters, and thus the initial data routine itself may never finish. Instead, recommended practice is to let Lorene output data into files, and then read those into ET at the beginning of a run.
+The ET contains three routines that can read in publicly available data generated by the
+{\tt Lorene} code \cite{loreneweb,Gourgoulhon:2000nn}, though it does not currently include
+the capability of generating such data from scratch. For a number of reasons, such functionality
+is not truly required; in particular, {\tt Lorene} is a serial code and there is no time-savings
+at all to call it as an ET initial data generator. Also, it is not guaranteed to be convergent
+for an arbitrary set of parameters, and thus the initial data routine itself may never finish.
+Instead, recommended practice is to let Lorene output data into files, and then read those
+into ET at the beginning of a run.
-Lorene uses a multigrid spectral approach to solve the conformal thin sandwich equations for binary initial configurations, and a single-grid spectral method for rotating stars. For binaries, five elliptical equations for the shift, lapse, and conformal factor are written down, and the source terms are divided into pieces that are attributed to each of the two objects. Matter source terms are ideal for this split, since they are compactly supported, while extrinsic curvature source terms are spatially extended, but with sufficiently rapid falloff at large radii to yield convergent solutions. Around each object, a set of nested spheroidal sub-domains is constructed extending to cover all of space, with the outermost domain incorporating a compactification to allow it to extend to spatial infinity. Within each of the nested subdomains, fields are decomposed into Chebyshev modes radially and spherical harmonics in the angular directions, with elliptic equation solving reduced to a matrix problem. The nested sub-domains are not required to be perfectly spherical, and indeed one may modify the outer boundaries of each to cover any convex shape. For neutron stars, this allows one to map the surface of a particular subdomain to the NS surface, eliminating Gibbs error at the surface. For BHs, excision boundary conditions are imposed at the horizon. To read a field solution describing a boundary onto a grid, one must incorporate the data from both of the domains at every point.
+Lorene uses a multigrid spectral approach to solve the conformal thin sandwich equations
+for binary initial configurations, and a single-grid spectral method for rotating stars.
+For binaries, five elliptical equations for the shift, lapse, and conformal factor are written down,
+and the source terms are divided into pieces that are attributed to each of the two objects.
+Matter source terms are ideal for this split, since they are compactly supported, while extrinsic
+curvature source terms are spatially extended, but with sufficiently rapid falloff at large radii
+to yield convergent solutions. Around each object, a set of nested spheroidal sub-domains
+is constructed extending to cover all of space, with the outermost domain incorporating a
+compactification to allow it to extend to spatial infinity. Within each of the nested subdomains,
+fields are decomposed into Chebyshev modes radially and spherical harmonics in the angular
+directions, with elliptic equation solving reduced to a matrix problem. The nested sub-domains
+are not required to be perfectly spherical, and indeed one may modify the outer boundaries
+of each to cover any convex shape. For neutron stars, this allows one to map the surface
+of a particular subdomain to the NS surface, eliminating Gibbs error at the surface.
+For BHs, excision boundary conditions are imposed at the horizon. To read a field solution
+describing a boundary onto a grid, one must incorporate the data from both of the domains
+at every point.
\begin{figure}
\label{fig:Lorene_coordinates}
\centering\includegraphics[width=0.5\textwidth]{Lorene_Grid}\\
- \caption{Example of a Lorene multi-domain coordinate system for binary initial data. The outermost, compactified domain extending to spatial infinity is not shown.}
+ \caption{Example of a Lorene multi-domain coordinate system for binary initial data.
+The outermost, compactified domain extending to spatial infinity is not shown.}
\end{figure}
\codename{Meudon\_Bin\_BH} can read in binary black hole
-initial data described in \cite{???}, while \codename{Meudon\_Bin\_NS} handles binary NS data from \cite{???}.
+initial data described in \cite{???}, while \codename{Meudon\_Bin\_NS} handles
+binary NS data from \cite{???}.
\codename{Meudon\_Mag\_NS} may be used to read in magnetized isolated neutron star data \cite{???}.
\codename{TOVSolver} for Tolman-Oppenheimer-Volkov type neutron star initial data,
