[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 37)
cott at tapir.caltech.edu
cott at tapir.caltech.edu
Sat Mar 26 10:14:54 CDT 2011
User: cott
Date: 2011/03/26 10:14 AM
Added:
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widetext.sty
/local_bibtex/
ott_references.bib
Modified:
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ET.tex
Log:
* bunch of changes; copied in some text from another paper that needs massaging
File Changes:
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===================================================================
--- ET.tex 2011-03-22 15:00:14 UTC (rev 36)
+++ ET.tex 2011-03-26 15:14:54 UTC (rev 37)
@@ -16,6 +16,7 @@
\usepackage[hyphens]{url}
\usepackage{wrapfig}
\usepackage{multicol}
+\usepackage{widetext}
\usepackage[bookmarks, bookmarksopen, bookmarksnumbered]{hyperref}
\usepackage[all]{hypcap}
@@ -388,6 +389,7 @@
supercomputers to run the same simulation there.
\subsection{Kranc\pages{1 Ian}}
+\label{sec:kranc}
Kranc\cite{Husa:2004ip,kranc04,krancweb} is a Mathematica application which converts a high-level
continuum description of a PDE into a highly optimised module for
@@ -542,13 +544,289 @@
the \codename{TwoPunctures} module can also be used to construct neutron star
black hole binary initial data, when being coupled with \codename{TOVSolver}.
-\subsection{Evolution Methods\pages{4 Christian}}
+\subsection{Spacetime Curvature and Hydrodynamics Evolution}
+\todo{Christian in charge}
-\paragraph{Spacetime}
+In the following, we assume that the reader is familiar with the
+basics of numerical relativity and GR hydrodynamics. Detailed
+introductions to numerical relativity have recently been given by
+Alcubierre~\cite{alcubierre:08} and Baumgarte \& Shapiro~\cite{baumgarte:10}.
+GR hydrodynamics has been reviewed by Font~\cite{font:08}.
-\paragraph{Relativistic Matter}
+\subsubsection{Spacetime Curvature Evolution} The Einstein Toolkit
+curvature evolution code \codename{McLachlan}~\cite{brown:09} is
+auto-generated from tensor equations via \codename{Kranc}
+(Sec.~\ref{sec:kranc}). It implements the Einstein equations in a
+$3+1$ split as a Cauchy initial boundary value problem
+\cite{york:79}. For this, the Baumgarte-Shapiro-Shibata-Nakamura
+(BSSN) conformal-tracefree reformulation
+\cite{shibata:95,baumgarte:95,alcubierre:00} of the original
+Arnowitt-Deser-Misner formalism~\cite{adm:62} is employed.
+This leads to the following set of evolved variables:
+\todo{continue editing here}
+
+\begin{eqnarray}
+ \phi & := & \log \left[ \frac{1}{12} \det \gamma_{ij} \right]
+ \\
+ \tilde\gamma_{ij} & := & e^{-4\phi}\; \gamma_{ij}
+ \\
+ K & := & g^{ij} K_{ij}
+ \\
+ \tilde A_{ij} & := & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K
+ \right]
+ \\
+ \tilde\Gamma^i & := & \tilde\gamma^{jk} \tilde\Gamma^i_{jk} .
+\end{eqnarray}
+Our exact evolution equations are as described by Eqs.~(3) to (10) of
+\cite{ES-Brown2007b}, which we list here for completeness:
+\begin{widetext}
+\begin{eqnarray}
+ \partial_0 \alpha & = & -\alpha^2 f(\alpha, \phi, x^\mu) (K -
+ K_0(x^\mu))
+ \\
+ \partial_0 K & = & -e^{-4\phi} \left[ \tilde{D}^i \tilde{D}_i \alpha
+ + 2 \partial_i \phi \cdot \tilde{D}^i \alpha \right] + \alpha
+ \left( \tilde{A}^{ij} \tilde{A}_{ij} + \frac{1}{3} K^2 \right) - \alpha S
+ \\
+ \partial_0 \beta^i & = & \alpha^2 G(\alpha,\phi,x^\mu) B^i
+ \\
+ \partial_0 B^i & = & e^{-4\phi} H(\alpha,\phi,x^\mu)
+ \partial_0\tilde{\Gamma}^i - \eta^i(B^i,\alpha,x^\mu)
+ \\
+ \partial_0 \phi & = & -\frac{\alpha}{6}\, K +
+ \frac{1}{6}\partial_k\beta^k
+ \\
+ \partial_0 \tilde{\gamma}_{ij} & = & -2\alpha\tilde{A}_{ij}
+ + 2\tilde{\gamma}_{k(i}\partial_{j)}\beta^k
+ - \frac{2}{3}\tilde{\gamma}_{ij}\partial_k\beta^k
+ \\
+ \partial_0 \tilde{A}_{ij} & = & e^{-4\phi}\left[
+ \alpha\tilde{R}_{ij} + \alpha R^\phi_{ij} - \tilde{D}_i\tilde{D}_j\alpha
+ + 4\partial_{(i}\phi\cdot\tilde{D}_{j)}\alpha\right]^{TF}
+ \nonumber\\
+ & & {} + \alpha K\tilde{A}_{ij} - 2\alpha\tilde{A}_{ik}\tilde{A}^k_{\; j}
+ + 2\tilde{A}_{k(i}\partial_{j)}\beta^k
+ - \frac{2}{3}\tilde{A}_{ij}\partial_k\beta^k
+ - \alpha e^{-4\phi} \hat{S}_{ij}
+ \\
+ \partial_0\tilde{\Gamma}^i & = &
+ \tilde{\gamma}^{kl}\partial_k\partial_l\beta^i
+ + \frac{1}{3} \tilde{\gamma}^{ij}\partial_j\partial_k\beta^k
+ + \partial_k\tilde{\gamma}^{kj} \cdot \partial_j\beta^i
+ - \frac{2}{3}\partial_k\tilde{\gamma}^{ki} \cdot \partial_j\beta^j
+ \nonumber\\
+ & & {} - 2\tilde{A}^{ij}\partial_j\alpha
+ + 2\alpha\left[ (m-1)\partial_k\tilde{A}^{ki} - \frac{2m}{3}\tilde{D}^i K
+ + m(\tilde{\Gamma}^i_{\; kl}\tilde{A}^{kl} + 6\tilde{A}^{ij}\partial_j\phi)
+ \right] - S^i
+\end{eqnarray}
+\end{widetext}
+with the momentum constraint damping constant set to $m=1$. The stress
+energy tensor $T_{\mu\nu}$ is incorporated via the projections
+\begin{eqnarray}
+ \rho & := & \frac{1}{\alpha^2} \left( T_{00} - 2 \beta^i T_{0i} +
+ \beta^i \beta^j T^{ij} \right)
+ \\
+ S & := & g^{ij} T_{ij}
+ \\
+ S_i & := & - \frac{1}{\alpha} \left( T_{0i} - \beta^j T_{ij} \right) .
+\end{eqnarray}
+We have introduced the notation $\partial_0 = \partial_t -
+\beta^j\partial_j$. All quantities with a tilde $\tilde{~}$ refer to
+the conformal 3-metric $\tilde{\gamma}_{ij}$, which is used to
+raise and lower indices. In particular, $\tilde{D}_i$ and
+$\tilde{\Gamma}^k_{ij}$ refer to the covariant derivative and the
+Christoffel symbols with respect to $\tilde{\gamma}_{ij}$. The
+expression $[ \cdots ]^{TF}$ denotes the trace-free part of the
+expression inside the parentheses, and we define the Ricci tensor
+contributions
+\begin{widetext}
+\begin{eqnarray}
+\tilde{R}_{ij}
+ &=& -\frac{1}{2} \tilde{\gamma}^{kl}\partial_k\partial_l\tilde{\gamma}_{ij}
+ + \tilde{\gamma}_{k(i}\partial_{j)}\tilde{\Gamma}^k
+ - \tilde{\Gamma}_{(ij)k}\partial_l\tilde{\gamma}^{lk}
+ + \tilde{\gamma}^{ls}\left( 2\tilde{\Gamma}^k_{\; l(i}\tilde{\Gamma}_{j)ks}
+ + \tilde{\Gamma}^k_{\; is}\tilde{\Gamma}_{klj} \right)
+\\
+R^\phi_{ij} &=& -2\tilde{D}_i\tilde{D}_j\phi
+ - 2\tilde{\gamma}_{ij} \tilde{D}^k\tilde{D}_k\phi
+ + 4\tilde{D}_i\phi\, \tilde{D}_j\phi
+ - 4\tilde{\gamma}_{ij}\tilde{D}^k\phi\, \tilde{D}_k\phi .
+\end{eqnarray}
+\end{widetext}
+This is a so-called $\phi$-variant of BSSN. The evolved gauge
+variables are lapse $\alpha$, shift $\beta^i$, and a quantity $B^i$
+related to the time derivative of the shift. The gauge parameters $f$,
+$G$, $H$, and $\eta$ are determined by our choice of a $1+\log$
+slicing:
+\begin{eqnarray}
+ f(\alpha,\phi,x^\mu) & := & 2/\alpha
+ \\
+ K_0(x^\mu) & := & 0
+\end{eqnarray}
+and $\Gamma$-driver shift condition:
+\begin{eqnarray}
+ G(\alpha,\phi,x^\mu) & := & (3/4)\, \alpha^{-2}
+ \\
+ H(\alpha,\phi,x^\mu) & := & \exp\{4\phi\}
+ \\\label{eq:eta}
+ \eta(B^i,\alpha,x^\mu) & := & (1/2)\, B^i q(r) .
+\end{eqnarray}
+The expression $q(r)$ attenuates the $\Gamma$-driver depending on the
+radius as described below.
+
+The $\Gamma$-driver shift condition is symmetry-seeking,
+driving the shift $\beta^i$ to a state that renders the conformal
+connection functions $\tilde\Gamma^i$
+stationary. Of course, such a stationary state cannot be achieved
+while the metric is evolving, but in a stationary spacetime the time
+evolution of the shift
+$\beta^i$ and thus that of the spatial coordinates $x^i$ will be exponentially
+damped. This damping time scale is set by the gauge parameter $\eta$
+(see Eq.~\ref{eq:eta}) which has dimension $1/T$ (inverse time).
+As described, e.g., in \cite{Muller:2009jx, ES-Schnetter2010a}, this
+time scale may need to be adapted in different regions of the domain
+to avoid spurious high-frequency behavior in regions that otherwise
+evolve only very slowly, e.g., far away from the source.
+
+Here we use the simple damping mechanism described in Eq.~(12) of
+\cite{ES-Schnetter2010a}, which is defined as
+\begin{eqnarray}
+ \label{eq:varying-simple}
+ q(r) & := & \left\{
+ \begin{array}{llll}
+ 1 & \mathrm{for} & r \le R & \textrm{(near the origin)}
+ \\
+ R/r & \mathrm{for} & r \ge R & \textrm{(far away)}
+ \end{array}
+ \right.
+\end{eqnarray}
+with a constant $R$ defining the transition radius between the
+interior, where $q\approx1$, and the exterior, where $q$ falls off as
+$1/r$. Eq.~\ref{eq:eta} describes how $q$ appears in the gauge
+parameters. In this paper we use $R=250\,M_\odot$ ($R =
+369.2\,\mathrm{km}$).
+
+We implement the above BSSN equations and gauge conditions in the
+{\tt McLachlan} code \cite{ES-Brown2007b, ES-mclachlanweb} which is
+freely available as part of the {\tt EinsteinToolkit}. {\tt McLachlan} is
+auto-generated from the definition of the variables and equations in the
+{\tt Mathematica} format by the {\tt Kranc} code generator \cite{kranc04,
+ Husa:2004ip, krancweb}. {\tt Kranc} is a suite of {\tt Mathematica} packages
+comprising a computer algebra toolbox for numerical relativists. {\tt Kranc}
+can be used as a ``rapid prototyping'' system for physicists or
+mathematicians handling complex systems of partial differential
+equations, and through integration into the {\tt Cactus} framework one can
+also produce efficient production codes.
+
+We use fourth-order accurate finite differencing for the spacetime
+variables and add a fifth-order Kreiss-Oliger dissipation term to
+remove high frequency noise. We use a fourth-order Runge-Kutta time
+integrator for all evolved variables.
+
+\subsubsection{Initial Conditions}
+
+We set up our initial condition from the ADM variables $g_{ij}$,
+$K_{ij}$, lapse $\alpha$, and shift $\beta^i$, as provided by the
+initial data discussed in Sec.~\ref{sec:initial_models}. From these we
+calculate the BSSN quantities via their definition, setting $B^i=0$,
+and using cubic extrapolation for $\tilde\Gamma^i$ at the outer
+boundary. This extrapolation is necessary since the $\tilde\Gamma^i$ are
+calculated from derivatives of the metric, and one cannot use centered
+finite differencing stencils near the outer boundary. We assume that
+one could instead also use one-sided derivatives to calculate
+$\tilde\Gamma^i$ on the boundary.
+
+The extrapolation stencils distinguish between points on the faces,
+edges, and corners of the grid. Points on the faces are extrapolated
+via stencils perpendicular to that face, while points on the edges and
+corners are extrapolated with stencils aligned with the average of the
+normals of the adjoining faces. For example, points on the $(+x,+y)$
+edge are extrapolated in the $(1,1,0)$ direction, while points in the
+$(+x,+y+z)$ corner are extrapolated in the $(1,1,1)$ direction. Since
+several layers of boundary points have to be filled for higher order
+schemes (e.g., three layers for a fourth order scheme), we proceed
+outwards starting from the innermost layer. Each subsequent layer is
+then defined via the points in the interior and the previously
+calculated layers.
+
+\subsubsection{Boundary Conditions}
+
+During time evolution, we apply a Sommerfeld-type radiative boundary
+condition to all components of the evolved BSSN variables as described
+in \cite{Alcubierre2000}. The main feature of this boundary condition
+is that it assumes approximate spherical symmetry of the solution,
+while applying the actual boundary condition on the boundary of a
+cubic grid where the face normals are not aligned with the radial
+direction. This boundary condition defines the right hand side
+of the BSSN state vector on the outer boundary, which is then
+integrated in time as well, so that the boundary and interior are
+calculated with the same order of accuracy.
+
+The main part of the boundary condition assumes that we have an
+outgoing radial wave with some speed $v_0$:
+\begin{eqnarray}
+ X & = & X_0 + \frac{u(r - v_0 t)}{r}
+\end{eqnarray}
+where $X$ is any of the tensor components of evolved variables, $X_0$
+the value at infinity, and $u$ a spherically symmetric perturbation.
+Both $X_0$ and $v_0$ depend on the particular variable and have to be
+specified. This implies the following differential equation:
+\begin{eqnarray}
+ \partial_t X & = & - v^i \partial_i X - v_0\, \frac{X - X_0}{r}\,,
+\end{eqnarray}
+where $v^i = v_0\, x^i/r$. The spatial derivatives $\partial_i$ are
+evaluated using centered finite differencing where possible, and
+one-sided finite differencing elsewhere. We use second order stencils
+in our implementation.
+
+In addition to this main part, we also account for those parts of the
+solution that do not behave as a pure wave, e.g., Coulomb type terms
+caused by infall of the coordinate lines. We assume that these parts
+decay with a certain power $p$ of the radius. We implement this by
+considering the radial derivative of the source term above, and
+extrapolating according to this power-law decay.
+
+Given a source term $(\partial_t X)$, we define the corrected source
+term $(\partial_t X)^*$ via
+\begin{eqnarray}
+ (\partial_t X)^* & = & (\partial_t X) + \left( \frac{r}{r - n^i
+ \partial_i r} \right)^p\; n^i \partial_i (\partial_t X)\,,
+\end{eqnarray}
+where $n^i$ is the normal vector of the corresponding boundary face.
+The spatial derivatives $\partial_i$ are evaluated by comparing
+neighbouring grid points, corresponding to a second-order stencil
+evaluated in the middle between the two neighbouring grid points. We
+assume a second-order decay, i.e., we choose $p=2$.
+
+As with the initial conditions above, this boundary condition is
+evaluated on several layers of grid points, starting from the
+innermost layer. Both the extrapolation and radiative boundary
+condition algorithms are implemented in the publicly available
+\texttt{NewRad} component of the Einstein Toolkit.
+
+This boundary condition is only a coarse approximation of the actual
+decay behavior of the BSSN state vector, and it does not capture the
+correct behavior of the evolved variables. However, we observe that
+this boundary condition leads to stable evolutions if applied
+sufficiently far from the source. Errors introduced at the boundary
+(both errors in the geometry and constraint violations) propagate
+inwards with the speed of light \cite{ES-Brown2007b}. Gauge changes
+introduced by the boundary condition, which are physically not
+observable, propagate faster, with a speed up to $\sqrt{2}$ for our
+gauge conditions.
+
+
+
+
+
+\subsubsection{Hydrodynamics: \codename{GRHydro}}
+
+
+
\subsection{Analysis\pages{4 Tanja}}
Any simulated system may have many applicable quantities for analysis which
@@ -696,6 +974,6 @@
\bibliographystyle{amsplain-url}
-\bibliography{manifest/einsteintoolkit}
+\bibliography{manifest/einsteintoolkit,local_bibtex/ott_references}
\end{document}
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--- widetext.sty (rev 0)
+++ widetext.sty 2011-03-26 15:14:54 UTC (rev 37)
@@ -0,0 +1,86 @@
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{widetext}
+
+%% Mimics the widetext environment of revtex4 for any other class package
+%% Eg: article.cls
+%%
+%% Compiled by: Anjishnu Sarkar
+%%
+%% Advantages:
+%% *) Supports float (eg: figure) in two column format (Advantage over
+%% multicol package)
+%% *) One and twocolumn exist on the same page
+%% *) Flow of text shown via rule
+%% *) Equal height of text when in two column format
+%%
+%% Acknowledgment(s):
+%% 1. Instead of re-inventing the wheel, two packages (flushend, cuted) of
+%% the sttools bundle are used. The sttools bundle is available from CTAN.
+%% Lisence of these packages rests with their corresponding author.
+%% Any bug/problem with flushend and cuted should be forwarded to their
+%% corresponding package authors.
+%% 2. The idea of the rule came from the following latex community website
+%% http://www.latex-community.org/forum/viewtopic.php?f=5&t=2770
+%%
+%% This package just defines the widetext environment and the rules.
+%%
+%% Usage:
+%% \documentclass[a4paper,12pt,twocolumn]{article}
+%% \usepackage{widetext}
+%%
+%% \begin{document}
+%%
+%% Some text in twocolumn
+%%
+%% \begin{widetext}
+%% Text in onecolumn format.
+%% \end{widetext}
+%%
+%% Some more text in twocolumn
+%%
+%% \end{document}
+%%%%%%%%%%%%%%%%%%%%
+
+%% Package required for equal height while in 2 columns format
+\IfFileExists{flushend.sty}
+ {\RequirePackage{flushend}}
+ {\typeout{}
+ \typeout{Package widetext error: Install the flushend package which is
+ a part of sttools bundle. Available from CTAN.}
+ \typeout{}
+ \stop
+ }
+
+%% Package required for onecolumn and twocolumn to exist on the same page.
+%% and also required for widetext environment.
+\IfFileExists{cuted.sty}
+ {\RequirePackage{cuted}}
+ {\typeout{}
+ \typeout{Package widetext error: Install the cuted package which is
+ a part of sttools bundle. Available from CTAN.}
+ \typeout{}
+ \stop
+ }
+
+
+\newlength\@parindent
+\setlength\@parindent{\parindent}
+
+\if at twocolumn
+ \newenvironment{widetext}
+ {%
+ \begin{strip}
+ \rule{\dimexpr(0.5\textwidth-0.5\columnsep-0.4pt)}{0.4pt}%
+ \rule{0.4pt}{6pt}
+ \par %\vspace{6pt}
+ \parindent \@parindent
+ }%
+ {%
+ \par
+ \hfill\rule[-6pt]{0.4pt}{6.4pt}%
+ \rule{\dimexpr(0.5\textwidth-0.5\columnsep-1pt)}{0.4pt}
+ \end{strip}
+ }
+\else
+ \newenvironment{widetext}{}{}
+\fi
Directory: /local_bibtex/
=========================
File [added]: ott_references.bib
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--- local_bibtex/ott_references.bib (rev 0)
+++ local_bibtex/ott_references.bib 2011-03-26 15:14:54 UTC (rev 37)
@@ -0,0 +1,104 @@
+ at incollection{york:79,
+ author = "York, James W.",
+ title = "Kinematics and Dynamics of General Relativity",
+ pages = "83--126",
+ editor = "Larry L. Smarr",
+ booktitle = "Sources of Gravitational Radiation",
+ publisher = "Cambridge University Press",
+ address = "Cambridge, UK",
+ year = 1979,
+ snote = "Proceedings of the Battelle Seattle Workshop,
+ 24 July -- 4 August, 1978",
+}
+
+ at Book{adm:62,
+author = {R.~Arnowitt and S.~Deser and C.~Misner},
+title = {Dynamics of General Relativity, in "Gravitation: An Introduction to Current Research"},
+publisher = {Wiley, New York},
+year = {1962},
+}
+
+ at article{shibata:95,
+ key = {Shibata95},
+ author = {Masaru Shibata and Takashi Nakamura},
+ journal = {Phys. Rev. D},
+ title = {Evolution of three-dimensional gravitational waves:
+ {H}armonic slicing case},
+ volume = {52},
+ year = {1995},
+ pages = {5428}
+}
+
+ at article{baumgarte:95,
+ key = {Baumgarte99},
+ author = {Thomas W. Baumgarte and Stuart L. Shapiro},
+ title = {On the numerical integration of {E}instein's field
+ equations},
+ journal = {Phys. Rev. D},
+ year = 1999,
+ volume = 59,
+ pages = 024007,
+}
+
+ at article{alcubierre:00,
+ author = {M. Alcubierre and B. Br\"{u}gmann and T. Dramlitsch and J. A. Font
+ and Philippos Papadopoulos and E. Seidel and N. Stergioulas and R. Takahashi},
+ title = {Towards a stable numerical evolution of strongly gravitating
+ systems in general relativity: The conformal treatments},
+ journal = {Phys. Rev. D},
+ volume = {62},
+ year = 2000,
+ pages = 044034,
+}
+
+
+ at article{alcubierre:03a,
+ author = {Miguel Alcubierre and Bernd Br\"ugmann and Peter Diener and
+ Michael Koppitz and Denis Pollney and Edward Seidel and Ryoji Takahashi},
+ title = "Gauge conditions for long-term numerical black hole
+ evolutions without excision",
+ journal = {Phys. Rev. D},
+ volume = {67},
+ pages = {084023},
+ year = 2003,
+}
+
+ at BOOK{alcubierre:08,
+ author = {{Alcubierre}, M.},
+ title = "{Introduction to 3+1 Numerical Relativity}",
+booktitle = {Introduction to 3+1 Numerical Relativity},
+ year = 2008,
+ editor = "{Alcubierre, M.}",
+publisher = {Oxford University Press},
+}
+
+ at BOOK{baumgarte:10,
+ author = {{Baumgarte}, T.~W. and {Shapiro}, S.~L.},
+ title = "{Numerical Relativity: Solving Einstein's Equations on the Computer}",
+ year = 2010,
+ publisher = "Cambridge University Press",
+}
+
+
+ at ARTICLE{font:08,
+ author = {{Font}, J.~A.},
+ title = "{Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity}",
+ journal = {Liv. Rev. Rel.},
+ keywords = {Hydrodynamics, Relativistic hydrodynamics, Numerical relativity, Magnetohydrodynamics},
+ year = 2008,
+ month = sep,
+ volume = 11,
+ pages = {7},
+}
+
+ at Article{brown:09,
+ author = {David Brown and Peter Diener and Olivier Sarbach and
+ Erik Schnetter and Manuel Tiglio},
+ title = {Turduckening black holes: an analytical and
+ computational study},
+ journal = {Phys. Rev. D},
+ year = 2009,
+ volume = 79,
+ pages = 044023,
+}
+
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