User: bmundim
Date: 2011/04/18 01:20 AM
Modified:
/
ET.tex
Log:
Add TOVSolver section; add references; minor edits throughout
the section (by Josh/Bruno).
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +95 -45
===================================================================
--- ET.tex 2011-04-18 02:44:28 UTC (rev 61)
+++ ET.tex 2011-04-18 06:20:06 UTC (rev 62)
@@ -716,7 +716,7 @@
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}),
+quantities defined in the Base modules (see section~\ref{sec:base_modules}),
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.
@@ -734,8 +734,12 @@
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).
+`moving puncture' gauge conditions (see, e.g., \cite{York:1998hy,Etienne:???}
+and other works).
+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{Simple Vacuum initial data}
Vacuum spacetime tests can be provided by \codename{IDConstraintViolate}, which
@@ -754,24 +758,25 @@
\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
+by \codename{GRHydro\_InitData}. This 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
+have been widely used throughout the field to evaluate a diverse set of solvers
+\cite{Marti:1999wi}.
+Conservative to primitive variable conversion and vice versa are also 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.
+(see Sec.~\ref{sec:GRHydro} for more on this). Along similar lines, the
+\codename{Hydro\_InitExcision} module sets up masks for different kinds of excised
+regions, convenient for hydrodynamics tests.
\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:???}.
+\codename{TwoPunctures} code, described originally in \cite{Ansorg:2004ds}, which solves
+the binary puncture equations for a pair of black holes \cite{Brandt:1997tf}.
To do so, one assumes the extrinsic curvature for each black hole corresponds to
-the Bowen-York form \cite{Bowen:???},
+the Bowen-York form \cite{Bowen:1980yu},
\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)
@@ -781,7 +786,7 @@
$\gamma^{ij}$ 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 ($g_{ij}=\psi^4\gamma_{ij}$):
+may be written as an elliptic 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*}
@@ -814,7 +819,7 @@
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.
+method to achieve rapid solutions and to avoid ill-conditioning of the spectral matrix.
\begin{figure}
\centering\includegraphics[width=0.5\textwidth]{TwoPunctures_grid_BHNS}\\
@@ -824,32 +829,36 @@
\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{Lorene:web,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 its iterative steps. 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~\cite{York:1998hy}, and a single-grid
+spectral method for rotating stars. For binaries, five elliptic 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}
\centering\includegraphics[width=0.5\textwidth]{Lorene_Grid}\\
@@ -858,22 +867,62 @@
\label{fig:Lorene_coordinates}
\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{???}.
- \codename{Meudon\_Mag\_NS} may be used to read in magnetized isolated neutron star data \cite{???}.
+ \codename{Meudon\_Bin\_BH} can read in binary black hole initial data described
+in \cite{Grandclement:2001ed}, while \codename{Meudon\_Bin\_NS}
+handles binary NS data from \cite{Gourgoulhon:2000nn}. \codename{Meudon\_Mag\_NS}
+may be used to read in magnetized isolated neutron star data \cite{Lorene:web}.
\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}.
+black hole binary initial data, when being coupled with \codename{TOVSolver}
+module.
-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{TOVSolver}
+The \codename{TOVSolver} routine in the ET solves the standard TOV equations
+\cite{Tolman:1939jz,Oppenheimer:1939ne} expressed using the Schwarzschild (areal)
+radius $r$ in the interior of a spherically symmetric star in hydrostatic equilibrium,
+\begin{eqnarray*}
+ \label{eq:TOViso}
+ \frac{d P}{d r} & = & -(\mu + P) \frac{m + 4\pi r^3 P}{r(r - 2m)}, \\
+%
+ \frac{d m}{d r} & = & 4 \pi r^2 \mu, \\
+%
+ \frac{d \phi}{d r} & = & \frac{m + 4\pi r^3 P}{r(r -
+ 2m)}
+\end{eqnarray*}
+where $m$ is the gravitational mass inside a sphere of radius $r$, and
+$\phi$ the logarithm of the lapse. It also supplies the analytically known
+solution in the exterior,
+\begin{eqnarray*}
+ \label{eq:TOVexterior}
+ P & = & {\tt TOV\_atmosphere}, \\
+ m & = & M, \\
+ \phi & = &\dfrac{1}{2} \log(1-2M / r)
+\end{eqnarray*}
+where {\tt TOV\_atmosphere} is a parameter used to define the density of the
+atmosphere in the exterior. Since the isotropic radius $\bar{r}$, is the more
+commonly preferred choice to initiate dynamical calculations, the code then
+transforms all variables into isotropic coordinates, integrating the radius
+conversion formula
+\begin{equation}
+\label{eq:rbar}
+\frac{d (\log(\bar{r} / r))}{\partial r} = \frac{r^{1/2} - (r-2m)^{1/2}}{r(r-2m)^{1/2}} \ .
+\end{equation}
+subject to the boundary condition that in the exterior,
+\begin{eqnarray*}
+\bar{r} &=& \dfrac{1}{2}\left(\sqrt{r^2-2Mr}+r -M\right) \\
+r&=&\bar{r}\left(1+\dfrac{M}{2\bar{r}}\right)^2 \ .
+\end{eqnarray*}
+handling with some care the potentially singular terms that appear at the origin.
+To facilitate the construction of stars in more complicated dynamical configurations,
+one may also apply a uniform velocity to the neutron star, though this does not affect
+the ODE solution nor the resulting density profile.
+
+
\subsection{Equations of States}\pages{1 Christian}
\subsection{Spacetime Curvature and Hydrodynamics Evolution}
@@ -1164,6 +1213,7 @@
\subsubsection{Hydrodynamics: \codename{GRHydro}}
+\label{sec:GRHydro}
