User: cott Date: 2011/03/26 10:14 AM

Added: / widetext.sty /local_bibtex/ ott_references.bib

Modified: / ET.tex

Log: * bunch of changes; copied in some text from another paper that needs massaging

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File [modified]: ET.tex Delta lines: +282 -4 =================================================================== --- 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}

File [added]: widetext.sty Delta lines: +86 -0 =================================================================== --- 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@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 Delta lines: +104 -0 =================================================================== --- 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 @@ +@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", +} + +@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}, +} + +@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} +} + +@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, +} + +@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, +} + + +@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, +} + +@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}, +} + +@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", +} + + +@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}, +} + +@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, +} +